<?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="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="armstrng.html#NM5">Full-family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:98"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="armstrng.html#R5" target="_self">is_generator-set_of</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:98"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="armstrng.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E4:98"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="armstrng.html#E3:98"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E5:98"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:98"><i><font color="Green">E56</font></i></a>, <a class="ref" href="armstrng.html#D26" target="_self">Def26</a></i>;<br/>
<a NAME="E6:98"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:98"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E9:98"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E8:98"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K4">INT</a>  as  <a href="relat_1.html#V2">V12</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E12:98"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0;<br/><b>thus </b><a NAME="E1:98_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E12:98"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:98_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E3:98_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="armstrng.html#E12:98"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E4:98_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E5:98_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="numbers.html#T17">NUMBERS:17</a></i>;<br/><b>hence </b><a NAME="E6:98_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E12:98"><i><font color="Green">E60</font></i></a>, <a class="txt" href="armstrng.html#E2:98_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:98"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0 is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2"/>
(  not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">(C1)</a> or <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">(C1)</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1"><b>suppose </b></a><a NAME="E1:98_2_1"/><i><font color="Green">E63</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">(C1)</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E2:98_2_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:98_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:98_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E13:98"><i><font color="Green">E61</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> ;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #);<br/><b>take </b>
 <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)
;<br/><b>thus </b><a NAME="E4:98_2_1"/>
( the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds the <a href="armstrng.html#U2" target="_self">Domains</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a>  ) )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #);<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #), B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) : <font color="Olive">b<sub>1</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Olive">b<sub>2</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_2_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:98_2_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E4:98_2_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:98"><i><font color="Green">E58</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) ;<br/><a NAME="E6:98_2_1_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">c<sub>16</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self">ARMSTRNG:def 8</a><br/>

<a NAME="E1:98_2_1_2_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:98_2_1_2_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>15</sub></font> implies <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E7:98_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #), B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) : <font color="Olive">b<sub>1</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Olive">b<sub>2</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_1_2_1"><i><font color="Green">E64</font></i></a></i>;<br/><b>hence </b><a NAME="E8:98_2_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:98_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)
 ;<br/><a NAME="E3:98_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #), B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) : <font color="Olive">b<sub>1</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Olive">b<sub>2</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)<b> such that </b><br/><a NAME="E5:98_2_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E6:98_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">c<sub>14</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)
 ;<br/><a NAME="E7:98_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_1"><i><font color="Green">E63</font></i></a></i>;<br/><b>hence </b><a NAME="E8:98_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:98"><i><font color="Green">E58</font></i></a>, <a class="txt" href="armstrng.html#E5:98_2_1_2"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:98_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2"><b>suppose </b></a><a NAME="E1:98_2_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">(C1)</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="armstrng.html#E8:98"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><a NAME="E3:98_2_2"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>7</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) );<br/><div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) );<br/><a NAME="E1:98_2_2_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:98_2_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_2_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> or  not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:98_2_2_1_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="margrel1.html#K7">FALSE</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  ;<br/><b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/><b>thus </b><a NAME="E3:98_2_2_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_1_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="margrel1.html#D13">MARGREL1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:98_2_2_1_1_1_2"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="margrel1.html#K8">TRUE</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a>  ;<br/><b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/><b>thus </b><a NAME="E3:98_2_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_1_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="margrel1.html#D14">MARGREL1:def 14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E3:98_2_2_1"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:98_2_2_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="armstrng.html#E1:98_2_2_1"><i><font color="Green">E66</font></i></a>);<br/></i><a NAME="E5:98_2_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E6:98_2_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:98_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E5:98_2_2_1"><i><font color="Green">E69</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>7</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  ;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><b>thus </b><a NAME="E9:98_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_1"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E6:98_2_2"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="armstrng.html#E4:98_2_2"><i><font color="Green">E65</font></i></a>);<br/></i><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = 0;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><i><font color="Green">E67</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
<font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="armstrng.html#T5" target="_self">Th5</a></i>
<br/>.= 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="binari_3.html#T26">BINARI_3:26</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> = 
0
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="binari_3.html#T7">BINARI_3:7</a></i>
;<br/></div><b>end;</b></div><a NAME="E8:98_2_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E14:98"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E9:98_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_2_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> </span>}</span> 
 ;<br/>

<a NAME="E2:98_2_2_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:98_2_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2"><i><font color="Green">E67</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #);<br/><b>take </b>
 <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)
;<br/><b>thus </b><a NAME="E11:98_2_2"/>
( the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds the <a href="armstrng.html#U2" target="_self">Domains</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a>  ) )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #);<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #), B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) : <font color="Olive">b<sub>1</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Olive">b<sub>2</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_2_2_4_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:98_2_2_4_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E4:98_2_2_4_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:98"><i><font color="Green">E58</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) ;<br/><a NAME="E6:98_2_2_4_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">c<sub>18</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self">ARMSTRNG:def 8</a><br/>

<b>assume </b><a NAME="E1:98_2_2_4_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<a NAME="E2:98_2_2_4_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E4:98_2_2_4_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E5:98_2_2_4_1_1"/><i><font color="Green">E72</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>
<a NAME="E6:98_2_2_4_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E8:98_2_2_4_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E9:98_2_2_4_1_1"/><i><font color="Green">E74</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:98_2_2_4_1_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:98_2_2_4_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:98_2_2_4_1_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:98_2_2_4_1_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<i><font color="Green">E77</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>20</sub></font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_4_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_1_1_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:98_2_2_4_1_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E14:98_2_2_4_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:98_2_2_4_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 ;<br/><a NAME="E3:98_2_2_4_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>;<br/><a NAME="E5:98_2_2_4_1_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><a NAME="E6:98_2_2_4_1_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>7</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_1_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:98_2_2_4_1_1_2_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:98_2_2_4_1_1_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_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="E1:98_2_2_4_1_1_2_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> or  not <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:98_2_2_4_1_1_2_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/><div class="add"><a NAME="E2:98_2_2_4_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:98"><i><font color="Green">E59</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:98_2_2_4_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_4_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="armstrng.html#E1:98_2_2_4_1_1_2_1_1_1"><i><font color="Green">E84</font></i></a></i>;<br/><i><font color="Green">E86</font></i>: <font color="Maroon">c<sub>31</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font> = 
<font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E5:98_2_2_4_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_1_1_2_1_1_1"><i><font color="Green">E85</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_4_1_1_2_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="margrel1.html#T49">MARGREL1:49</a>, <a class="ref" href="margrel1.html#D13">MARGREL1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_4_1_1_2_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="margrel1.html#T49">MARGREL1:49</a>, <a class="ref" href="margrel1.html#D13">MARGREL1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:98_2_2_4_1_1_2_1_1_2"/><i><font color="Green">E84</font></i>: 
not <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:98_2_2_4_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>33</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:98_2_2_4_1_1_2_1_1_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E3:98_2_2_4_1_1_2_1_1_2_1"/><i><font color="Green">E86</font></i>: 
not <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_1_1_2_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E85</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>7</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  ;<br/>
<a NAME="E6:98_2_2_4_1_1_2_1_1_2_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E87</font></i>: <font color="Maroon">c<sub>35</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font> = 
<font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E86</font></i></a></i>
;<br/>
<i><font color="Green">E88</font></i>: <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E9:98_2_2_4_1_1_2_1_1_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_4_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="armstrng.html#E11:98_2_2_4_1_1"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E10:98_2_2_4_1_1_2_1_1_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="armstrng.html#E13:98_2_2_4_1_1"><i><font color="Green">E76</font></i></a></i>;<br/>
<i><font color="Green">E91</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E7:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="margrel1.html#T50">MARGREL1:50</a>, <a class="ref" href="margrel1.html#D14">MARGREL1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="armstrng.html#E10:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="binari_3.html#T2">BINARI_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E7:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="margrel1.html#T50">MARGREL1:50</a>, <a class="ref" href="margrel1.html#D14">MARGREL1:def 14</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="armstrng.html#E11:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E9:98_2_2_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>then </b><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> = 
 <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="armstrng.html#E13:98_2_2_4_1_1"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_4_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="armstrng.html#E11:98_2_2_4_1_1"><i><font color="Green">E75</font></i></a></i>
;<br/><b>hence </b><a NAME="E11:98_2_2_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:98_2_2_4_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:98_2_2_4_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #), B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) : <font color="Olive">b<sub>1</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Olive">b<sub>2</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_1"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E8:98_2_2_4_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:98_2_2_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)
 ;<br/><a NAME="E3:98_2_2_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #), B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) : <font color="Olive">b<sub>1</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Olive">b<sub>2</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)<b> such that </b><br/><a NAME="E5:98_2_2_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E6:98_2_2_4"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">c<sub>16</sub></font>, <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:98_2_2_4_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:98_2_2_4_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E3:98_2_2_4_2"/><i><font color="Green">E72</font></i>: 
not <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_2"><i><font color="Green">E70</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:98_2_2_4_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; not <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_2"><i><font color="Green">E73</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:98_2_2_4_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E10:98_2_2_4_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:98"><i><font color="Green">E59</font></i></a>, <a class="txt" href="armstrng.html#E1:98_2_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E11:98_2_2_4_2"/><i><font color="Green">E76</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:98_2_2_4_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:98_2_2_4_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E15:98_2_2_4_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E11:98_2_2_4_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="armstrng.html#E14:98_2_2_4_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  =  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:98_2_2_4_2"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  =  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:98_2_2_4_2"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E20:98_2_2_4_2"/><i><font color="Green">E81</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E12:98"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:98_2_2_4_2_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:98_2_2_4_2_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E17:98_2_2_4_2"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E4:98_2_2_4_2_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E5:98_2_2_4_2_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="numbers.html#T17">NUMBERS:17</a></i>;<br/><b>hence </b><a NAME="E6:98_2_2_4_2_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E20:98_2_2_4_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E23:98_2_2_4_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E17:98_2_2_4_2"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E24:98_2_2_4_2"/><i><font color="Green">E83</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E12:98"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:98_2_2_4_2_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:98_2_2_4_2_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E19:98_2_2_4_2"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E4:98_2_2_4_2_2"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E5:98_2_2_4_2_2"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="numbers.html#T17">NUMBERS:17</a></i>;<br/><b>hence </b><a NAME="E6:98_2_2_4_2_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E24:98_2_2_4_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E27:98_2_2_4_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E19:98_2_2_4_2"><i><font color="Green">E80</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2_3"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:98_2_2_4_2_3"/><i><font color="Green">E85</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E20:98_2_2_4_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="armstrng.html#E24:98_2_2_4_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:98_2_2_4_2_3"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 ;<br/><a NAME="E3:98_2_2_4_2_3"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_2_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0;<br/><b>set </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>;<br/><i><font color="Green">E87</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E17:98_2_2_4_2"><i><font color="Green">E79</font></i></a></i>
;<br/><a NAME="E6:98_2_2_4_2_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E19:98_2_2_4_2"><i><font color="Green">E80</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>7</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  ;<br/><b>set </b><font color="Maroon">c<sub>32</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>33</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2_3_2"><b>now </b></a><div class="add"><b>thus </b> <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>
;<br/><b>let </b><font color="Maroon">c<sub>34</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:98_2_2_4_2_3_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:98_2_2_4_2_3_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>34</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="armstrng.html#E2:98_2_2_4_2_3_2"><i><font color="Green">E89</font></i></a></i>;<br/><i><font color="Green">E91</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0 = 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="binari_3.html#T26">BINARI_3:26</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
;<br/><a NAME="E6:98_2_2_4_2_3_2"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>then </b><i><font color="Green">E92</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="armstrng.html#E5:98_2_2_4_2_3_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green">E93</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="armstrng.html#E7:98_2_2_4_2_3_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="margrel1.html#T49">MARGREL1:49</a>, <a class="ref" href="margrel1.html#D13">MARGREL1:def 13</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_2_3_2_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_2_4_2_3_2_5"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>35</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_2_3_2_5_1"><b>suppose </b></a><a NAME="E1:98_2_2_4_2_3_2_5_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/><div class="add"><a NAME="E2:98_2_2_4_2_3_2_5_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>then </b><i><font color="Green">E95</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E14:98_2_2_4_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="armstrng.html#E15:98_2_2_4_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="armstrng.html#E1:98_2_2_4_2_3_2_5_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="armstrng.html#T8" target="_self">Th8</a>, <a class="ref" href="margrel1.html#D13">MARGREL1:def 13</a></i>
;<br/><a NAME="E4:98_2_2_4_2_3_2_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:98_2_2_4_2_3_2_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_2_3_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2_5_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="margrel1.html#T49">MARGREL1:49</a>, <a class="ref" href="margrel1.html#D13">MARGREL1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_2_3_2_5_2"><b>suppose </b></a><a NAME="E1:98_2_2_4_2_3_2_5_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/><div class="add"><a NAME="E2:98_2_2_4_2_3_2_5_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>then </b><i><font color="Green">E95</font></i>: <font color="Maroon">c<sub>31</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> = 
<font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="armstrng.html#E1:98_2_2_4_2_3_2_5_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="armstrng.html#E10:98_2_2_4_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_2_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="armstrng.html#E1:98_2_2_4_2_3_2_5_2"><i><font color="Green">E94</font></i></a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2_5_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="margrel1.html#T49">MARGREL1:49</a>, <a class="ref" href="margrel1.html#D13">MARGREL1:def 13</a></i>
;<br/><b>hence </b><a NAME="E5:98_2_2_4_2_3_2_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_2_3_2"><i><font color="Green">E93</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:98_2_2_4_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:98_2_2_4_2_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_2_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><a NAME="E29:98_2_2_4_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E30:98_2_2_4_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4"><i><font color="Green">E69</font></i></a>, <a class="txt" href="armstrng.html#E23:98_2_2_4_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="armstrng.html#E27:98_2_2_4_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self">Def8</a></i>;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0;<br/><b>set </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>;<br/><i><font color="Green">E86</font></i>:  <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E17:98_2_2_4_2"><i><font color="Green">E79</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="armstrng.html#E30:98_2_2_4_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E19:98_2_2_4_2"><i><font color="Green">E80</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>7</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  ;<br/><a NAME="E33:98_2_2_4_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>then </b><i><font color="Green">E87</font></i>: <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="armstrng.html#E10:98_2_2_4_2"><i><font color="Green">E75</font></i></a></i>
;<br/><i><font color="Green">E88</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0 = 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="binari_3.html#T26">BINARI_3:26</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
;<br/><a NAME="E36:98_2_2_4_2"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>then </b><i><font color="Green">E89</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="armstrng.html#E35:98_2_2_4_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green">E90</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E34:98_2_2_4_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="margrel1.html#T50">MARGREL1:50</a>, <a class="ref" href="margrel1.html#D14">MARGREL1:def 14</a></i>
;<br/><i><font color="Green">E91</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="armstrng.html#K3" target="_self">'&amp;'</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E34:98_2_2_4_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="margrel1.html#T50">MARGREL1:50</a>, <a class="ref" href="margrel1.html#D14">MARGREL1:def 14</a></i>
;<br/><a NAME="E40:98_2_2_4_2"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="armstrng.html#E14:98_2_2_4_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="armstrng.html#E15:98_2_2_4_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="armstrng.html#T8" target="_self">Th8</a></i>
;<br/><b>hence </b><a NAME="E42:98_2_2_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E31:98_2_2_4_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="armstrng.html#E37:98_2_2_4_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="armstrng.html#E38:98_2_2_4_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="armstrng.html#E39:98_2_2_4_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="binari_3.html#T2">BINARI_3:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:98_2_2_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:98"><i><font color="Green">E58</font></i></a>, <a class="txt" href="armstrng.html#E5:98_2_2_4"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:98_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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