<?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">PA</font>, <font color="Olive" title="b2">PB</font>, <font color="Olive" title="b3">PC</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  <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="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Olive" title="b3">PC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">PB</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Olive" title="b3">PC</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">PA</font>, <font color="Maroon" title="c3">PB</font>, <font color="Maroon" title="c4">PC</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">PB</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font></span>)</span></span><br/>



<b>consider </b><font color="Maroon" title="c5">PD</font>, <font color="Maroon" title="c6">PE</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E2:20"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c5">PD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</font>
 <b>and </b><br/><a NAME="E3:20"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c6">PE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PB</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font>
 ;<br/>
<a NAME="E4:20"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> holds <br/> ex <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font> &amp; <font color="Olive" title="b1">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">v</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">u</font> be    <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"> ( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> implies  ex <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font> &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">v</font> ) )</span><br/>

<b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font> &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">v</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c8">u1</font>, <font color="Maroon" title="c9">u2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PD</font>
 <b>and </b><br/><a NAME="E4:20_1_1"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c9">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">PC</font>
 <b>and </b><br/><a NAME="E5:20_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">u2</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/>
<b>consider </b><font color="Maroon" title="c10">u3</font>, <font color="Maroon" title="c11">u4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:20_1_1"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c10">u3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 <b>and </b><br/><a NAME="E8:20_1_1"/><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon" title="c11">u4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 <b>and </b><br/><a NAME="E9:20_1_1"/><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c8">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">u4</font>
 <b>by </b><i><a class="txt" href="partit1.html#E2:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="partit1.html#E3:20_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">v</font>, <font color="Maroon" title="c13">v1</font>, <font color="Maroon" title="c14">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:20_1_1"/><i><font color="Green" title="E25">A11</font></i>: 
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u4</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">u2</font>
 <b>and </b><br/><a NAME="E12:20_1_1"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c14">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u3</font>
 <b>and </b><br/><a NAME="E13:20_1_1"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">u3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">u4</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">u2</font>
 ;<br/>
<a NAME="E14:20_1_1"/><i><font color="Green" title="E28">A14</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c13">v1</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E11:20_1_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="partit1.html#E12:20_1_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="partit1.html#E13:20_1_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E15:20_1_1"/><i><font color="Green" title="E29">A15</font></i>: 
<font color="Maroon" title="c13">v1</font> <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">PB</font>,<font color="Maroon" title="c4">PC</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E4:20_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="partit1.html#E8:20_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="partit1.html#E11:20_1_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E16:20_1_1"/><i><font color="Green" title="E30">A16</font></i>: 
not <font color="Maroon" title="c7">u</font> <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="partit1.html#E1:20_1_1"><i><font color="Green" title="E18">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="E17:20_1_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c13">v1</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E5:20_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="partit1.html#E9:20_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="partit1.html#E11:20_1_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E18:20_1_1"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="partit1.html#E16:20_1_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E19:20_1_1"/><b>then </b>
not <font color="Maroon" title="c13">v1</font> <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="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:20_1_1"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">PB</font>,<font color="Maroon" title="c4">PC</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="txt" href="partit1.html#E15:20_1_1"><i><font color="Green" title="E29">A15</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="E21:20_1_1"/><b>then </b><i><font color="Green" title="E31">A17</font></i>: 
<font color="Maroon" title="c12">v</font> <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="c2">PA</font>,<font color="Maroon" title="c6">PE</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E3:20"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="partit1.html#E7:20_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="partit1.html#E12:20_1_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="partit1.html#E14:20_1_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font> &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">v</font> )</span><br/>

<b>thus </b><a NAME="E22:20_1_1"/>
( <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font> &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">v</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E5:20_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="partit1.html#E9:20_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="partit1.html#E13:20_1_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="partit1.html#E16:20_1_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="partit1.html#E21:20_1_1"><i><font color="Green" title="E31">A17</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>
<b>hence </b><a NAME="E2:20_1"/>
<font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</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="E5:20"/>
<font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_2"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font> holds <br/> ex <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> &amp; <font color="Olive" title="b1">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">v</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">u</font> be    <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"> ( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font> implies  ex <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">v</font> ) )</span><br/>

<b>assume </b><a NAME="E1:20_2_1"/><i><font color="Green" title="E18">A18</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">v</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c8">u1</font>, <font color="Maroon" title="c9">u2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_2_1"/><i><font color="Green" title="E19">A19</font></i>: 
<font color="Maroon" title="c8">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 <b>and </b><br/><a NAME="E4:20_2_1"/><i><font color="Green" title="E20">A20</font></i>: 
<font color="Maroon" title="c9">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">PE</font>
 <b>and </b><br/><a NAME="E5:20_2_1"/><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">u2</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/>
<b>consider </b><font color="Maroon" title="c10">u3</font>, <font color="Maroon" title="c11">u4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:20_2_1"/><i><font color="Green" title="E22">A22</font></i>: 
<font color="Maroon" title="c10">u3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 <b>and </b><br/><a NAME="E8:20_2_1"/><i><font color="Green" title="E23">A23</font></i>: 
<font color="Maroon" title="c11">u4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">PC</font>
 <b>and </b><br/><a NAME="E9:20_2_1"/><i><font color="Green" title="E24">A24</font></i>: 
<font color="Maroon" title="c9">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u3</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">u4</font>
 <b>by </b><i><a class="txt" href="partit1.html#E3:20"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="partit1.html#E4:20_2_1"><i><font color="Green" title="E20">A20</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E10:20_2_1"/><i><font color="Green" title="E25">A25</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">u1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">u3</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">u4</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E5:20_2_1"><i><font color="Green" title="E21">A21</font></i></a>, <a class="txt" href="partit1.html#E9:20_2_1"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">v</font>, <font color="Maroon" title="c13">v1</font>, <font color="Maroon" title="c14">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:20_2_1"/><i><font color="Green" title="E26">A26</font></i>: 
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">u3</font>
 <b>and </b><br/><a NAME="E13:20_2_1"/>
<font color="Maroon" title="c14">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u4</font>
 <b>and </b><br/><a NAME="E14:20_2_1"/><i><font color="Green" title="E27">A27</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">u1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">u3</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">u4</font>
 ;<br/>
<a NAME="E15:20_2_1"/><i><font color="Green" title="E28">A28</font></i>: 
<font color="Maroon" title="c13">v1</font> <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="c2">PA</font>,<font color="Maroon" title="c3">PB</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E3:20_2_1"><i><font color="Green" title="E19">A19</font></i></a>, <a class="txt" href="partit1.html#E7:20_2_1"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="partit1.html#E12:20_2_1"><i><font color="Green" title="E26">A26</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E16:20_2_1"/><i><font color="Green" title="E29">A29</font></i>: 
not <font color="Maroon" title="c7">u</font> <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="partit1.html#E1:20_2_1"><i><font color="Green" title="E18">A18</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="E17:20_2_1"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="partit1.html#E10:20_2_1"><i><font color="Green" title="E25">A25</font></i></a>, <a class="txt" href="partit1.html#E12:20_2_1"><i><font color="Green" title="E26">A26</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:20_2_1"/><b>then </b>
not <font color="Maroon" title="c13">v1</font> <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="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E19:20_2_1"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">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="txt" href="partit1.html#E15:20_2_1"><i><font color="Green" title="E28">A28</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="E20:20_2_1"/><b>then </b><i><font color="Green" title="E30">A30</font></i>: 
<font color="Maroon" title="c12">v</font> <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="c5">PD</font>,<font color="Maroon" title="c4">PC</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E2:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="partit1.html#E8:20_2_1"><i><font color="Green" title="E23">A23</font></i></a>, <a class="txt" href="partit1.html#E12:20_2_1"><i><font color="Green" title="E26">A26</font></i></a>, <a class="txt" href="partit1.html#E14:20_2_1"><i><font color="Green" title="E27">A27</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">v</font> )</span><br/>

<b>thus </b><a NAME="E21:20_2_1"/>
( <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">v</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E10:20_2_1"><i><font color="Green" title="E25">A25</font></i></a>, <a class="txt" href="partit1.html#E14:20_2_1"><i><font color="Green" title="E27">A27</font></i></a>, <a class="txt" href="partit1.html#E16:20_2_1"><i><font color="Green" title="E29">A29</font></i></a>, <a class="txt" href="partit1.html#E20:20_2_1"><i><font color="Green" title="E30">A30</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>
<b>hence </b><a NAME="E2:20_2"/>
<font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c6">PE</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Maroon" title="c5">PD</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</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="E6:20"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c3">PB</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">PB</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">PC</font></span>)</span>
 <b>by </b><i><a class="txt" href="partit1.html#E2:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="partit1.html#E3:20"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="partit1.html#E4:20"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="partit1.html#T5" target="_self" title="PARTIT1:th.5">Th5</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>
