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

<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><br/> for <font color="Olive" title="b2">PA</font>, <font color="Olive" title="b3">PB</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Olive" title="b1">z</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Olive" title="b3">PB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b3">PB</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">PA</font>, <font color="Olive" title="b2">PB</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Olive" title="b2">PB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Olive" title="b1">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Olive" title="b2">PB</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">PA</font>, <font color="Maroon" title="c4">PB</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font></span>)</span></span><br/>





<a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</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" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font></span>)</span></span><br/>

<a NAME="E2:1_1"/><i><font color="Green" title="E4">A3</font></i>: 
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font>
 
<b>by </b><i><a class="ref" href="bvfunc11.html#T3" title="BVFUNC11:th.3">BVFUNC11:3</a></i>;<br/>
<a NAME="E3:1_1"/>
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font>
 
<b>by </b><i><a class="ref" href="bvfunc11.html#T3" title="BVFUNC11:th.3">BVFUNC11:3</a></i>;<br/>
<b>hence </b><a NAME="E4:1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E2:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</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" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:1_2"/><i><font color="Green" title="E3">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> ;<br/>
<a NAME="E3:1_2"/><i><font color="Green" title="E4">A5</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font> )
 
<b>by </b><i><a class="txt" href="#E1:1_2"><i><font color="Green" title="E3">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:1_2"/><i><font color="Green" title="E5">A6</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">PA</font> &amp;  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">PA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PA</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E5:1_2"/><i><font color="Green" title="E6">A7</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">PB</font> &amp;  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">PB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">PB</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E6:1_2"/><i><font color="Green" title="E7">A8</font></i>: 
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">PA</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font>
 
<b>by </b><i><a class="txt" href="#E3:1_2"><i><font color="Green" title="E4">A5</font></i></a>, <a class="txt" href="#E4:1_2"><i><font color="Green" title="E5">A6</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E7:1_2"/><i><font color="Green" title="E8">A9</font></i>: 
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">PB</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font>
 
<b>by </b><i><a class="txt" href="#E3:1_2"><i><font color="Green" title="E4">A5</font></i></a>, <a class="txt" href="#E5:1_2"><i><font color="Green" title="E6">A7</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E8:1_2"/><i><font color="Green" title="E9">A10</font></i>: 
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font> &amp;  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PA</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E9:1_2"/><i><font color="Green" title="E10">A11</font></i>: 
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font> &amp;  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">PB</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E10:1_2"/><i><font color="Green" title="E11">A12</font></i>: 
<font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c3">PA</font>,<font color="Maroon" title="c4">PB</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="partit1.html#D4" title="PARTIT1:def.4">PARTIT1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">Z</font> =  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span>;<br/>
<a NAME="E11:1_2"/>
(  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c3">PA</font>,<font color="Maroon" title="c4">PB</font> &amp; not  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E10:1_2"><i><font color="Green" title="E11">A12</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">X</font>, <font color="Maroon" title="c9">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:1_2"/><i><font color="Green" title="E12">A13</font></i>: 
( <font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PA</font> &amp; <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">PB</font> &amp;  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">Y</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E14:1_2"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">Y</font>
 
<b>by </b><i><a class="txt" href="#E13:1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E15:1_2"/><b>then </b><i><font color="Green" title="E13">A14</font></i>: 
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> &amp; <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E16:1_2"/><b>then </b>
<font color="Maroon" title="c8">X</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font>
 
<b>by </b><i><a class="txt" href="#E8:1_2"><i><font color="Green" title="E9">A10</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E17:1_2"/><b>then </b><i><font color="Green" title="E14">A15</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font>
 
<b>by </b><i><a class="txt" href="#E13:1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E18:1_2"/>
<font color="Maroon" title="c9">Y</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font>
 
<b>by </b><i><a class="txt" href="#E9:1_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="txt" href="#E15:1_2"><i><font color="Green" title="E13">A14</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E19:1_2"/><b>then </b><i><font color="Green" title="E15">A16</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font>
 
<b>by </b><i><a class="txt" href="#E13:1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E20:1_2"/><i><font color="Green" title="E16">A17</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">PA</font>
 
<b>by </b><i><a class="txt" href="#E6:1_2"><i><font color="Green" title="E7">A8</font></i></a>, <a class="txt" href="#E17:1_2"><i><font color="Green" title="E14">A15</font></i></a>, <a class="ref" href="eqrel_1.html#T50" title="EQREL_1:th.50">EQREL_1:50</a></i>;<br/>
<a NAME="E21:1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">PB</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:1_2"><i><font color="Green" title="E5">A6</font></i></a>, <a class="txt" href="#E5:1_2"><i><font color="Green" title="E6">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E22:1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:1_2"><i><font color="Green" title="E8">A9</font></i></a>, <a class="txt" href="#E13:1_2"><i><font color="Green" title="E12">A13</font></i></a>, <a class="txt" href="#E19:1_2"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="#E20:1_2"><i><font color="Green" title="E16">A17</font></i></a>, <a class="ref" href="eqrel_1.html#T50" title="EQREL_1:th.50">EQREL_1:50</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PB</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
