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

<b>set </b><font color="Maroon">f</font> =  <a href="margrel1.html#K7">FALSE</a> ;<br/>
<b>set </b><font color="Maroon">t</font> =  <a href="margrel1.html#K8">TRUE</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">z1</font>, <font color="Olive">z2</font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">a<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  holds  <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <span class="p2">(<span class="default"><font color="Olive">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Olive">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Olive">z1</font>,<font color="Olive">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Olive">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Olive">z2</font></span>)</span>;<br/>
<a NAME="E1:25"/><i><font color="Green">E29</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">z1</font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">z2</font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<a NAME="E1:25_1"/>
<span class="p1">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:25_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span>, <font color="Maroon">F</font> = <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> as   <a href="binarith.html#NM1">Tuple</a> of 1,<a href="margrel1.html#K6">BOOLEAN</a>  ;<br/>
<a NAME="E4:25_1"/><i><font color="Green">E41</font></i>: 
(  <a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </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">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> ) or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> ) or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> ) or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> ) )
 </span><b>by </b><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">E42</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <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" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a>, , </i>;<br/>
<b>hence </b><a NAME="E2:25_1_1_1_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 ;<br/><a NAME="E2:25_1_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
;<br/><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" target="_self">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z2</font></span>)</span>
 <b>by </b><i>, , , </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">E45</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_2"/><i><font color="Green">E46</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1"><b>proof </b></a><div class="add">

 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> = 
<a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
;<br/>
<b>hence </b><a NAME="E2:25_1_1_2_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 ;<br/><a NAME="E2:25_1_1_2_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">T</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
;<br/><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" target="_self">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z2</font></span>)</span>
 <b>by </b><i>, , , </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">E54</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_3"/><i><font color="Green">E55</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3_1"><b>proof </b></a><div class="add">

 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> = 
<a href="margrel1.html#K7">FALSE</a>  <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 

;<br/>
<b>hence </b><a NAME="E2:25_1_1_3_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_3_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 ;<br/><a NAME="E2:25_1_1_3_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">T</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
;<br/><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" target="_self">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z2</font></span>)</span>
 <b>by </b><i>, , , </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">E57</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_4"/><i><font color="Green">E58</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="hidden.html#R1">=</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" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a>, , </i>;<br/>
<b>hence </b> <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> = 
2 <a href="series_1.html#K3">to_power</a> 1
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="power.html#T30">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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_4_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 ;<br/><a NAME="E2:25_1_1_4_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><a NAME="E4:25_1_1_4"/><b>then </b>
<font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11" target="_self">*&gt;</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:25_1_1_4"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z2</font></span>)</span>
 <b>by </b><i>, , </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">E60</font></i>: 
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</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">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>let </b><font color="Maroon">z1</font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">z2</font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>consider </b><font color="Maroon">t1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">d1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E3:25_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<b>consider </b><font color="Maroon">t2</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">d2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E5:25_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<a NAME="E6:25_2"/>
(  <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d1</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d2</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon">C1</font> =  <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>, <font color="Maroon">C2</font> =  <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">C3</font> =  <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d1</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">C4</font> =  <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d2</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">C5</font> =  <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<i><font color="Green">E65</font></i>:  <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p5">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p0"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p0"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p5">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p0"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p0"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p5">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p0"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p0"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p5">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p0"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p0"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p5"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E9:25_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">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">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <font color="Maroon">d1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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">E67</font></i>: 
<font color="Maroon">d1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">C3</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <font color="Maroon">d2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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">E69</font></i>: 
<font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">C4</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2_1_1_1_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_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_1_1_1"/>
(  <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or  <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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_1_1_1"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">C5</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:25_2_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:25_2_2_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:25_2_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
 <b>by </b><i>, , , <a class="txt" href="binarith.html#E1:25"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T3" target="_self">Th3</a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_2_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E100</font></i>: 
 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">C5</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:25_2_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:25_2_2_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, , , <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><a NAME="E5:25_2_2_1_1_1_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="binarith.html#T3" target="_self">Th3</a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E6:25_2_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
 <b>by </b><i>, , <a class="txt" href="binarith.html#E1:25"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<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_1"/>
<font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
 ;<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">E103</font></i>: 
<font color="Maroon">d2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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">E104</font></i>: 
<font color="Maroon">C4</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or  <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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">E105</font></i>: 
 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">C5</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:25_2_2_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:25_2_2_1_1_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, , , <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><a NAME="E5:25_2_2_1_1_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="binarith.html#T3" target="_self">Th3</a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E6:25_2_2_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
 <b>by </b><i>, , <a class="txt" href="binarith.html#E1:25"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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">E108</font></i>: 
 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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">E109</font></i>: 
<font color="Maroon">C5</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><i><font color="Green">E110</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K1" target="_self">'or'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span> = 
<a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
;<br/><a NAME="E5:25_2_2_1_1_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="binarith.html#T3" target="_self">Th3</a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
<b>by </b><i><a class="ref" href="binarith.html#T2" target="_self">Th2</a>, , <a class="txt" href="binarith.html#E1:25"><i><font color="Green">E29</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">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">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">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">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">E111</font></i>: 
<font color="Maroon">d1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">C3</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <font color="Maroon">d2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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">E113</font></i>: 
<font color="Maroon">d2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">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">E114</font></i>: 
<font color="Maroon">C4</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or  <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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"/><i><font color="Green">E115</font></i>: 
 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:25_2_2_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">C5</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:25_2_2_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:25_2_2_1_1_2_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, , , <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><a NAME="E5:25_2_2_1_1_2_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="binarith.html#T3" target="_self">Th3</a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E6:25_2_2_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
 <b>by </b><i>, , <a class="txt" href="binarith.html#E1:25"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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">E118</font></i>: 
 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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">E119</font></i>: 
<font color="Maroon">C5</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><i><font color="Green">E120</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 

;<br/><span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span> = 
<a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
;<br/><a NAME="E5:25_2_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="binarith.html#T3" target="_self">Th3</a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
<b>by </b><i><a class="ref" href="binarith.html#T2" target="_self">Th2</a>, , <a class="txt" href="binarith.html#E1:25"><i><font color="Green">E29</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">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">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">E121</font></i>: 
<font color="Maroon">d2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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">E122</font></i>: 
<font color="Maroon">C4</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or  <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">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">E123</font></i>: 
 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">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">E124</font></i>: 
<font color="Maroon">C5</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><i><font color="Green">E125</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 

;<br/><span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a>, </i>
;<br/><a NAME="E5:25_2_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="binarith.html#T3" target="_self">Th3</a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
2 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
<b>by </b><i><a class="ref" href="binarith.html#T2" target="_self">Th2</a>, , <a class="txt" href="binarith.html#E1:25"><i><font color="Green">E29</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">E126</font></i>: 
 <a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><i><font color="Green">E127</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" target="_self">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d2</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 

;<br/><span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
<a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>
;<br/><a NAME="E4:25_2_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="binarith.html#E1:25"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><font color="Maroon">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
<b>by </b><i><a class="ref" href="binarith.html#T2" target="_self">Th2</a>, , , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</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">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">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">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">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">C2</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C5</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">t2</font></span>)</span> <a href="binarith.html#K10" target="_self">^</a> <span class="p3"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p5">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">t1</font>,<font color="Maroon">t2</font></span>)</span></span>)</span></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">t2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C2</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C1</font>
<b>by </b><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" target="_self">Absval</a> <span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">t2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C5</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
<b>by </b><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" target="_self">Absval</a> <font color="Maroon">t1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">t2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">t1</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C3</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">t2</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">C4</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">t1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">d1</font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <span class="p2">(<span class="default"><font color="Maroon">t2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p3"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" target="_self">Absval</a> <font color="Maroon">z2</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:25"/>
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1" target="_self">BINARITH:sch 1</a>(, );</i><br/>


</div>
