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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<a NAME="E1:8"/><i><font color="Green">E1</font></i>: 
 <a href="bvfunc_2.html#K7">Ex</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K24">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D10">BVFUNC_2:def 10</a></i>;<br/>
<a NAME="E2:8"/><i><font color="Green">E2</font></i>: 
 <a href="bvfunc_2.html#K7">Ex</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K24">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D10">BVFUNC_2:def 10</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bvfunc_1.html#D15">BVFUNC_1:def 15</a><br/>

<b>assume </b><a NAME="E3:8"/><i><font color="Green">E3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K9">'xor'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<i><font color="Green">E4</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bvfunc_1.html#K9">'xor'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D8">BVFUNC_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#D2">BINARITH:def 2</a></i>
;<br/>
<a NAME="E5:8"/><i><font color="Green">E5</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <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="E6:8"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E7:8"/><i><font color="Green">E7</font></i>: 
(  <a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K7">FALSE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K8">TRUE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green">E8</font></i>: 
<a href="margrel1.html#K7">FALSE</a>  <a href="margrel1.html#K12">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="margrel1.html#T49">MARGREL1:49</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_2"/>
( <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 </span><b>by </b><i><a class="txt" href="bvfunc_3.html#E3:8"><i><font color="Green">E3</font></i></a>, <a class="txt" href="bvfunc_3.html#E4:8"><i><font color="Green">E4</font></i></a>, <a class="txt" href="bvfunc_3.html#E5:8"><i><font color="Green">E5</font></i></a>, <a class="ref" href="binarith.html#T7">BINARITH:7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1"><b>suppose </b></a><a NAME="E1:8_2_1"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:8_2_1"/><b>then </b><i><font color="Green">E9</font></i>: 
(  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><a NAME="E3:8_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><a NAME="E4:8_2_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K24">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_3.html#E6:8"><i><font color="Green">E6</font></i></a>, <a class="ref" href="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/><a NAME="E5:8_2_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K24">B_SUP</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_3.html#E6:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="bvfunc_3.html#E2:8_2_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/><a NAME="E6:8_2_1"/><b>then </b><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_2.html#D10">BVFUNC_2:def 10</a></i>;<br/><i><font color="Green">E12</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D8">BVFUNC_1:def 8</a></i>
<br/>.= 
<a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="bvfunc_3.html#E1:8"><i><font color="Green">E1</font></i></a>, <a class="txt" href="bvfunc_3.html#E7:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="bvfunc_3.html#E8:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bvfunc_3.html#E4:8_2_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="bvfunc_3.html#E6:8_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="binarith.html#D2">BINARITH:def 2</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>
;<br/><a NAME="E8:8_2_1"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p4">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p4">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="bvfunc_1.html#K8">'or'</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7">BVFUNC_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="bvfunc_3.html#E7:8_2_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="bvfunc_3.html#E8:8_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>
<br/>.= 
<a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_2"><b>suppose </b></a><a NAME="E1:8_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:8_2_2"/><b>then </b><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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#T45">MARGREL1:45</a></i>;<br/><a NAME="E3:8_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><a NAME="E4:8_2_2"/><b>then </b><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K24">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_3.html#E6:8"><i><font color="Green">E6</font></i></a>, <a class="ref" href="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/><a NAME="E5:8_2_2"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K24">B_SUP</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_3.html#E6:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="bvfunc_3.html#E2:8_2_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/><a NAME="E6:8_2_2"/><b>then </b><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_2.html#D10">BVFUNC_2:def 10</a></i>;<br/><i><font color="Green">E12</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D8">BVFUNC_1:def 8</a></i>
<br/>.= 
<a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="bvfunc_3.html#E2:8"><i><font color="Green">E2</font></i></a>, <a class="txt" href="bvfunc_3.html#E7:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="bvfunc_3.html#E8:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bvfunc_3.html#E4:8_2_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="bvfunc_3.html#E6:8_2_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="binarith.html#D2">BINARITH:def 2</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>
;<br/><a NAME="E8:8_2_2"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p4">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p4">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="bvfunc_1.html#K8">'or'</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7">BVFUNC_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span>
<b>by </b><i><a class="txt" href="bvfunc_3.html#E7:8_2_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="bvfunc_3.html#E8:8_2_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p5">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K9">'xor'</a> <span class="p4">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
