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

<b>set </b><font color="Maroon" title="c1">f</font> =  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ;<br/>
<b>set </b><font color="Maroon" title="c2">t</font> =  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>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 $1,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  holds  <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Olive" title="b2">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> $1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Olive" title="b1">z1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Olive" title="b2">z2</font></span>)</span>;<br/>
<a NAME="E1:25"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z1</font>, <font color="Maroon" title="c4">z2</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of 1,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/>

<a NAME="E1:25_1"/><i><font color="Green" title="E18">A2</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="binarith.html#D5" target="_self" title="BINARITH:def.5">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">T</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>, <font color="Maroon" title="c6">F</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of 1,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ;<br/>
<a NAME="E3:25_1"/><i><font color="Green" title="E19">A3</font></i>: 
(  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c5">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="binarith.html#T41" target="_self" title="BINARITH:th.41">Th41</a>, <a class="ref" href="binarith.html#T42" target="_self" title="BINARITH:th.42">Th42</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1"/>
( ( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> ) or ( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> ) or ( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> ) or ( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> ) )
 </span><b>by </b><i><a class="ref" href="binarith.html#T40" target="_self" title="BINARITH:th.40">Th40</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_1_1_1"/>
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>;<br/>
<b>hence </b><a NAME="E2:25_1_1_1_1"/>
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_1_2"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c7">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> 1
 ;<br/><a NAME="E2:25_1_1_1_2"/><b>then </b><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>thus </b><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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="c4">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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#K5" title="BINARITH:func.5">carry</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E2:25_1_1_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="binarith.html#D5" target="_self" title="BINARITH:def.5">Def5</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_1_1"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c3">z1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c4">z2</font></span>)</span>
 <b>by </b><i><a class="txt" href="binarith.html#E3:25_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="binarith.html#E2:25_1_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="binarith.html#D8" target="_self" title="BINARITH:def.8">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2"/><i><font color="Green" title="E20">A8</font></i>: 
( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_2"/><i><font color="Green" title="E21">A9</font></i>: 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_1_2_1"/>
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>;<br/>
<b>hence </b><a NAME="E2:25_1_1_2_1"/>
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_2_2"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c7">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> 1
 ;<br/><a NAME="E2:25_1_1_2_2"/><b>then </b><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">T</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1_1_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_2_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1_1_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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="c4">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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#K5" title="BINARITH:func.5">carry</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E2:25_1_1_2_2"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="binarith.html#D5" target="_self" title="BINARITH:def.5">Def5</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_1_2"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c3">z1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c4">z2</font></span>)</span>
 <b>by </b><i><a class="txt" href="binarith.html#E3:25_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="binarith.html#E2:25_1_1_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="binarith.html#D8" target="_self" title="BINARITH:def.8">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_3"><b>suppose </b></a><a NAME="E1:25_1_1_3"/><i><font color="Green" title="E20">A12</font></i>: 
( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_3"/><i><font color="Green" title="E21">A13</font></i>: 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_1_3_1"/>
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>;<br/>
<b>hence </b><a NAME="E2:25_1_1_3_1"/>
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_3_2"/>
<font color="Maroon" title="c7">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> 1
 ;<br/><a NAME="E2:25_1_1_3_2"/><b>then </b><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c5">T</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1_1_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1_1_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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="c4">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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#K5" title="BINARITH:func.5">carry</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E2:25_1_1_3_2"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="binarith.html#D5" target="_self" title="BINARITH:def.5">Def5</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_1_3"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c3">z1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c4">z2</font></span>)</span>
 <b>by </b><i><a class="txt" href="binarith.html#E3:25_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="binarith.html#E2:25_1_1_3"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="binarith.html#D8" target="_self" title="BINARITH:def.8">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4"><b>suppose </b></a><a NAME="E1:25_1_1_4"/><i><font color="Green" title="E20">A15</font></i>: 
( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_4"/><i><font color="Green" title="E21">A16</font></i>: 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_1_4_1"/>
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_4"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>;<br/>
<b>hence </b> <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> = 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
;<br/><br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_4_2"/>
<font color="Maroon" title="c7">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> 1
 ;<br/><a NAME="E2:25_1_1_4_2"/><b>then </b><i><font color="Green" title="E22">A17</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1_1_4"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_1_1_4"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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="c4">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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#K5" title="BINARITH:func.5">carry</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E2:25_1_1_4_2"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="binarith.html#D5" target="_self" title="BINARITH:def.5">Def5</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E4:25_1_1_4"/><b>then </b>
<font color="Maroon" title="c3">z1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="binarith.html#D8" target="_self" title="BINARITH:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_4"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c4">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c3">z1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c4">z2</font></span>)</span>
 <b>by </b><i><a class="txt" href="binarith.html#E3:25_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="binarith.html#E1:25_1_1_4"><i><font color="Green" title="E20">A15</font></i></a>, <a class="txt" href="binarith.html#E2:25_1_1_4"><i><font color="Green" title="E21">A16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:25"/><i><font color="Green" title="E18">A18</font></i>: 
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green" title="E19">A19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c4">z1</font>, <font color="Maroon" title="c5">z2</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/>

<b>consider </b><font color="Maroon" title="c6">t1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> , <font color="Maroon" title="c7">d1</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> <b> such that </b><br/><a NAME="E3:25_2"/><i><font color="Green" title="E20">A20</font></i>: 
<font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137" title="FINSEQ_2:th.137">FINSEQ_2:137</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">t2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> , <font color="Maroon" title="c9">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> <b> such that </b><br/><a NAME="E5:25_2"/><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c5">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137" title="FINSEQ_2:th.137">FINSEQ_2:137</a></i>;<br/>
<a NAME="E6:25_2"/>
(  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c7">d1</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c9">d2</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">C1</font> =  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>, <font color="Maroon" title="c11">C2</font> =  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span>, <font color="Maroon" title="c12">C3</font> =  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c7">d1</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span>, <font color="Maroon" title="c13">C4</font> =  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c9">d2</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span>, <font color="Maroon" title="c14">C5</font> =  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<i><font color="Green" title="E22">A22</font></i>:  <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">z2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p0"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p0"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E3:25_2"><i><font color="Green" title="E20">A20</font></i></a>, <a class="txt" href="binarith.html#E5:25_2"><i><font color="Green" title="E21">A21</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"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p0"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p0"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p0"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p0"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p0"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p0"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">d1</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T44" target="_self" title="BINARITH:th.44">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T44" target="_self" title="BINARITH:th.44">Th44</a></i>
;<br/>
<a NAME="E9:25_2"/><i><font color="Green" title="E23">A23</font></i>: 
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_2_1_1"/>
( <font color="Maroon" title="c7">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  or <font color="Maroon" title="c7">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_1"/><i><font color="Green" title="E24">A24</font></i>: 
<font color="Maroon" title="c7">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E25">A25</font></i>: 
<font color="Maroon" title="c12">C3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_2_1_1_1_1_1"/>
( <font color="Maroon" title="c9">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  or <font color="Maroon" title="c9">d2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_1_1_1_1"/><i><font color="Green" title="E26">A26</font></i>: 
<font color="Maroon" title="c9">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">C4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:25_2_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 <b>by </b><i><a class="txt" href="binarith.html#E8:25_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_1_1_1_1"><i><font color="Green" title="E26">A26</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_1_1_1_2"/><i><font color="Green" title="E26">A27</font></i>: 
<font color="Maroon" title="c9">d2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A28</font></i>: 
<font color="Maroon" title="c13">C4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_2_1_1_1_1_1_2_1_1"/>
(  <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  or  <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E28">A29</font></i>: 
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">C5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:25_2_2_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 <b>by </b><i><a class="txt" href="binarith.html#E8:25_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E28">A29</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E28">A30</font></i>: 
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E29">A31</font></i>: 
<font color="Maroon" title="c14">C5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><i><font color="Green" title="E30">A32</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_1_1_1_2"><i><font color="Green" title="E26">A27</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E28">A30</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c9">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#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span> = 
<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_1_1_1_2"><i><font color="Green" title="E26">A27</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E28">A30</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
;<br/><a NAME="E5:25_2_2_1_1_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="binarith.html#E8:25_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="binarith.html#E3:25_2_2_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E30">A32</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
<b>by </b><i><a class="txt" href="binarith.html#E2:25_2_2_1_1_1"><i><font color="Green" title="E25">A25</font></i></a>, <a class="txt" href="binarith.html#E2:25_2_2_1_1_1_1_1_2"><i><font color="Green" title="E27">A28</font></i></a>, <a class="txt" href="binarith.html#E2:25_2_2_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E29">A31</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:25_2_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:25_2_2_1_1_1"/>
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_2"/><i><font color="Green" title="E24">A33</font></i>: 
<font color="Maroon" title="c7">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E25">A34</font></i>: 
<font color="Maroon" title="c12">C3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_2_1_1_2_1_1"/>
( <font color="Maroon" title="c9">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  or <font color="Maroon" title="c9">d2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_2_1_1_1"/><i><font color="Green" title="E26">A35</font></i>: 
<font color="Maroon" title="c9">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E27">A36</font></i>: 
<font color="Maroon" title="c13">C4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_2_1_1_2_1_1_1_1_1"/>
(  <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  or  <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_2_1_1_1_1_1_1"/>
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_2_2_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 <b>by </b><i><a class="txt" href="binarith.html#E8:25_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_1"><i><font color="Green" title="E26">A35</font></i></a>, <a class="txt" href="binarith.html#E2:25_2_2_1_1_2_1_1_1"><i><font color="Green" title="E27">A36</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E28">A37</font></i>: 
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E29">A38</font></i>: 
<font color="Maroon" title="c14">C5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><i><font color="Green" title="E30">A39</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2"><i><font color="Green" title="E24">A33</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_1"><i><font color="Green" title="E26">A35</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E28">A37</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c9">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#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span> = 
<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2"><i><font color="Green" title="E24">A33</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_1"><i><font color="Green" title="E26">A35</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E28">A37</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
;<br/><a NAME="E5:25_2_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="binarith.html#E8:25_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="binarith.html#E3:25_2_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E30">A39</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
<b>by </b><i><a class="txt" href="binarith.html#E2:25_2_2_1_1_2"><i><font color="Green" title="E25">A34</font></i></a>, <a class="txt" href="binarith.html#E2:25_2_2_1_1_2_1_1_1"><i><font color="Green" title="E27">A36</font></i></a>, <a class="txt" href="binarith.html#E2:25_2_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E29">A38</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:25_2_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_2_1_1_2"/><i><font color="Green" title="E26">A40</font></i>: 
<font color="Maroon" title="c9">d2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E27">A41</font></i>: 
<font color="Maroon" title="c13">C4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_2_1_1_2_1_1_2_1_1"/>
(  <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  or  <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E28">A42</font></i>: 
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E29">A43</font></i>: 
<font color="Maroon" title="c14">C5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><i><font color="Green" title="E30">A44</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2"><i><font color="Green" title="E24">A33</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_2"><i><font color="Green" title="E26">A40</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c9">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#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c9">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#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2"><i><font color="Green" title="E24">A33</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_2"><i><font color="Green" title="E26">A40</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E28">A42</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
;<br/><a NAME="E5:25_2_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="binarith.html#E8:25_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="binarith.html#E3:25_2_2_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E30">A44</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
<b>by </b><i><a class="txt" href="binarith.html#E2:25_2_2_1_1_2"><i><font color="Green" title="E25">A34</font></i></a>, <a class="txt" href="binarith.html#E2:25_2_2_1_1_2_1_1_2"><i><font color="Green" title="E27">A41</font></i></a>, <a class="txt" href="binarith.html#E2:25_2_2_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E29">A43</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E28">A45</font></i>: 
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><div class="add"><i><font color="Green" title="E29">A46</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2"><i><font color="Green" title="E24">A33</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_2"><i><font color="Green" title="E26">A40</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c9">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#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c9">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#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2"><i><font color="Green" title="E24">A33</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_2"><i><font color="Green" title="E26">A40</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E28">A45</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>
;<br/><a NAME="E4:25_2_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="binarith.html#E8:25_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="binarith.html#E2:25_2_2_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E29">A46</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
<b>by </b><i><a class="txt" href="binarith.html#E2:25_2_2_1_1_2"><i><font color="Green" title="E25">A34</font></i></a>, <a class="txt" href="binarith.html#E2:25_2_2_1_1_2_1_1_2"><i><font color="Green" title="E27">A41</font></i></a>, <a class="txt" href="binarith.html#E1:25_2_2_1_1_2_1_1_2_1_1_2"><i><font color="Green" title="E28">A45</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:25_2_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:25_2_2_1_1_2"/>
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_2_2"/>
<font color="Maroon" title="c11">C2</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">C5</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Maroon" title="c5">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c9">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p5">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c6">t1</font>,<font color="Maroon" title="c8">t2</font></span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">z2</font></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E3:25_2"><i><font color="Green" title="E20">A20</font></i></a>, <a class="txt" href="binarith.html#E5:25_2"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="binarith.html#T45" target="_self" title="BINARITH:th.45">Th45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Maroon" title="c8">t2</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c11">C2</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c10">C1</font>
<b>by </b><i><a class="ref" href="binarith.html#T46" target="_self" title="BINARITH:th.46">Th46</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#K7" title="BINARITH:func.7">Absval</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <font color="Maroon" title="c8">t2</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c14">C5</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
<b>by </b><i><a class="txt" href="binarith.html#E9:25_2"><i><font color="Green" title="E23">A23</font></i></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#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c6">t1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c8">t2</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font>
<b>by </b><i><a class="txt" href="binarith.html#E1:25_2"><i><font color="Green" title="E19">A19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c6">t1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c12">C3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c13">C4</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c6">t1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c7">d1</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t2</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c9">d2</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T46" target="_self" title="BINARITH:th.46">Th46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c4">z1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c5">z2</font></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E3:25_2"><i><font color="Green" title="E20">A20</font></i></a>, <a class="txt" href="binarith.html#E5:25_2"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="binarith.html#T46" target="_self" title="BINARITH:th.46">Th46</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:25"/>
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1" target="_self" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="binarith.html#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="binarith.html#E2:25"><i><font color="Green" title="E18">A18</font></i></a>);</i><br/>


</div>
