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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:1"/><i><font color="Green">E2</font></i>: 
 <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:1_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:1_1"/><i><font color="Green">E4</font></i>: 
 <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc11.html#T3">BVFUNC11:3</a></i>;<br/>
<a NAME="E3:1_1"/>
 <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc11.html#T3">BVFUNC11:3</a></i>;<br/>
<b>hence </b><a NAME="E4:1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E1:1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="bvfunc14.html#E2:1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:1_2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:1_2"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E1:1_2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:1_2"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E5:1_2"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E6:1_2"/><i><font color="Green">E7</font></i>: 
 <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E3:1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="bvfunc14.html#E4:1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E7:1_2"/><i><font color="Green">E8</font></i>: 
 <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E3:1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="bvfunc14.html#E5:1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E8:1_2"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E9:1_2"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E10:1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="partit1.html#D4">PARTIT1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E11:1_2"/>
(  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; not  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E10:1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:1_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E14:1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E13:1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E15:1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E8:1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E17:1_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E13:1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E18:1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E9:1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="bvfunc14.html#E15:1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E19:1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E13:1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E20:1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E6:1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="bvfunc14.html#E17:1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="t_1topsp.html#T9">T_1TOPSP:9</a></i>;<br/>
<a NAME="E21:1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E4:1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="bvfunc14.html#E5:1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E22:1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E7:1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bvfunc14.html#E13:1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="bvfunc14.html#E19:1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bvfunc14.html#E20:1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="t_1topsp.html#T9">T_1TOPSP:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K22">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
