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

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









<b>assume </b><a NAME="E1:18"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="bvfunc_2.html#V2">independent</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span>;<br/>
<a NAME="E2:18"/><i><font color="Green">E14</font></i>: 
 <a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_2.html#K2">'/\'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D7">BVFUNC_2:def 7</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green">E15</font></i>: 
 <a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_2.html#K2">'/\'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D7">BVFUNC_2:def 7</a></i>;<br/>
<a NAME="E4:18"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E5:18"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E1:18"><i><font color="Green">E13</font></i></a>, <a class="ref" href="partit_2.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E6:18"/><i><font color="Green">E17</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 <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> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) implies <span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( ex <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; not <span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) implies <span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E18</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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:18_1_1"/>
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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:18_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:18_1_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E2:18"><i><font color="Green">E14</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:18_1_1_1"/>
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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:18_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E3:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:18_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E5:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="partit_2.html#E2:18_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:18_1_1_1_1"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:18_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<a NAME="E7:18_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E6:18_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</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="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E9:18_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E3:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="partit_2.html#E6:18_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E10:18_1_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="partit_2.html#E1:18_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E11:18_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E2:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="partit_2.html#E5:18_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E12:18_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="partit_2.html#E10:18_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_1_1_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:18_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:18_1"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:18_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:18_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="partit_2.html#E3:18_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/>
<a NAME="E7:18_1"/><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E3:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="partit_2.html#E2:18_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E8:18_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E2:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="partit_2.html#E5:18_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E9:18_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E5:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="partit_2.html#E7:18_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:18_1"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E12:18_1"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<a NAME="E13:18_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="partit1.html#K4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2">'/\'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="bvfunc_2.html#K4">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="bvfunc_2.html#K4">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E11:18_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</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="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E15:18_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E2:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="partit_2.html#E12:18_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E16:18_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="partit_2.html#E3:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="partit_2.html#E11:18_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E17:18_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="partit_2.html#E6:18_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/>
<b>hence </b><a NAME="E18:18_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="partit_2.html#E15:18_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b> <a href="bvfunc_2.html#K6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="bvfunc_1.html#K23">B_INF</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9">BVFUNC_2:def 9</a></i>
<br/>.= 
 <a href="bvfunc_1.html#K23">B_INF</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9">BVFUNC_2:def 9</a></i>
<br/>.= 
 <a href="bvfunc_1.html#K23">B_INF</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K23">B_INF</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="partit_2.html#E6:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>
<br/>.= 
 <a href="bvfunc_1.html#K23">B_INF</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9">BVFUNC_2:def 9</a></i>
<br/>.= 
 <a href="bvfunc_2.html#K6">All</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9">BVFUNC_2:def 9</a></i>
;<br/>


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