<?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">G</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">G</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>  holds <br/> ex <font color="Olive" title="b3">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b3">X</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Olive" title="b1">G</font> )</span><br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>; <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">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Maroon" title="c2">G</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>  holds <br/> ex <font color="Olive" title="b2">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b2">X</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> )</span><br/><b>let </b><font color="Maroon" title="c3">y</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"> ( <font color="Maroon" title="c2">G</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>  implies  ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> ) )</span><br/>





<b>assume </b><a NAME="E1:6"/>
<font color="Maroon" title="c2">G</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> 
 ; <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">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c4">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g</font> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="bvfunc_2.html#E3:6"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/>
<a NAME="E5:6"/><i><font color="Green" title="E6">A2</font></i>: 
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> holds <br/>( <font color="Olive" title="b1">A</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>  &amp; ( for <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> or <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> or <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> ) ) ) ) )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E6:6"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 
;<br/>
<a NAME="E7:6"/><i><font color="Green" title="E8">A4</font></i>: 
for <font color="Olive" title="b1">d</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Maroon" title="c1">Y</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b1">d</font>
 
<b>by </b><i><a class="ref" href="partit1.html#T9" title="PARTIT1:th.9">PARTIT1:9</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/>$1 <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b1">d</font> ) );<br/>
<b>reconsider </b><font color="Maroon" title="c6">XX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span>}</span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font> <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c7">XX</font> = <font color="Maroon" title="c6">XX</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font> ;<br/>
<a NAME="E10:6"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">XX</font>
 
<b>by </b><i><a class="txt" href="bvfunc_2.html#E6:6"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="bvfunc_2.html#E7:6"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<a NAME="E11:6"/>
for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">XX</font> holds <br/><font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">X1</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="c8">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">XX</font> implies <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X1</font> )</span><br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon" title="c8">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">XX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X1</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c9">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X1</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Maroon" title="c9">X</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b1">d</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:6_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X1</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E3:6_1"><i><font color="Green" title="E10">A6</font></i></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="E12:6"/><b>then </b><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">XX</font>
 
<b>by </b><i><a class="txt" href="bvfunc_2.html#E10:6"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E13:6"/><b>then </b><i><font color="Green" title="E11">A8</font></i>: 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</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="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>
<b>take </b>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> &amp;  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> )</span><br/>

<a NAME="E14:6"/><i><font color="Green" title="E12">A9</font></i>: 
for <font color="Olive" title="b1">d</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/> <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">d</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"> ( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> implies  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c8">d</font> )</span><br/>

<b>assume </b><a NAME="E1:6_2"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c8">d</font></span><br/>

<a NAME="E2:6_2"/>
for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">XX</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c8">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">X1</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="c9">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">XX</font> implies <font color="Maroon" title="c9">X1</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c8">d</font> )</span><br/>

<b>assume </b><a NAME="E1:6_2_1"/>
<font color="Maroon" title="c9">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">XX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">X1</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c8">d</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:6_2_1"/><i><font color="Green" title="E14">A11</font></i>: 
( <font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">X1</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X</font> &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Maroon" title="c10">X</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b1">d</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:6_2_1"/>
<font color="Maroon" title="c9">X1</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c8">d</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:6_2"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="bvfunc_2.html#E3:6_2_1"><i><font color="Green" title="E14">A11</font></i></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="E3:6_2"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c8">d</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E13:6"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="partit1.html#T10" title="PARTIT1:th.10">PARTIT1:10</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><b>end;</b></div>
<a NAME="E15:6"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> &amp; ( for <font color="Olive" title="b2">d</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Olive" title="b1">e</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b2">d</font> ) holds <br/><font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">e</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="c8">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Maroon" title="c8">e</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b1">d</font> ) implies <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> )</span><br/>

<b>assume </b><a NAME="E1:6_3"/><i><font color="Green" title="E13">A12</font></i>: 
( <font color="Maroon" title="c8">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Maroon" title="c8">e</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b1">d</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font></span><br/>

<a NAME="E2:6_3"/><b>then </b>
<font color="Maroon" title="c8">e</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="txt" href="bvfunc_2.html#E3:6"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">Ad</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_3"/><i><font color="Green" title="E14">A13</font></i>: 
( <font color="Maroon" title="c9">Ad</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c9">Ad</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>  &amp; <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">Ad</font> )
 <b>by </b><i><a class="ref" href="partit1.html#D1" title="PARTIT1:def.1">PARTIT1:def 1</a></i>;<br/>
<a NAME="E5:6_3"/><i><font color="Green" title="E15">A14</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="txt" href="bvfunc_2.html#E5:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="bvfunc_2.html#E4:6_3"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_3_1"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e</font> or  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_1"><b>suppose </b></a><a NAME="E1:6_3_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font></span><br/><div class="add"><a NAME="E2:6_3_1_1"/><b>then </b><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">XX</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:6_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="bvfunc_2.html#E5:6_3"><i><font color="Green" title="E15">A14</font></i></a></i>;<br/><a NAME="E3:6_3_1_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">X1</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="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> or <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e</font> )</span><br/>

<b>assume </b><a NAME="E1:6_3_1_1_1"/>
<font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e</font></span><br/>

<a NAME="E2:6_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">XX</font>
 
<b>by </b><i><a class="txt" href="bvfunc_2.html#E10:6"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:6_3_1_1_1"/>
<font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E2:6_3_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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="E4:6_3_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:6_3"><i><font color="Green" title="E13">A12</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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_3_1_2"><b>suppose </b></a><a NAME="E1:6_3_1_2"/><i><font color="Green" title="E16">A16</font></i>: 
not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c8">e</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="bvfunc_2.html#E5:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="bvfunc_2.html#E4:6_3"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/><a NAME="E3:6_3_1_2"/>
<font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="ref" href="subset_1.html#D5" title="SUBSET_1:def.5">SUBSET_1:def 5</a></i>;<br/><a NAME="E4:6_3_1_2"/><b>then </b><i><font color="Green" title="E17">A17</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:6_3_1_2"><i><font color="Green" title="E16">A16</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="E5:6_3_1_2"/>
<font color="Maroon" title="c10">e</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>;<br/><a NAME="E6:6_3_1_2"/><b>then </b><i><font color="Green" title="E18">A18</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E7:6_3_1_2"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:6_3_1_2"><i><font color="Green" title="E16">A16</font></i></a></i>;<br/><a NAME="E8:6_3_1_2"/><b>then </b>
for <font color="Olive" title="b1">d</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b1">d</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:6_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="partit1.html#T12" title="PARTIT1:th.12">PARTIT1:12</a></i>;<br/><a NAME="E9:6_3_1_2"/><b>then </b><i><font color="Green" title="E19">A19</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">XX</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E4:6_3_1_2"><i><font color="Green" title="E17">A17</font></i></a></i>;<br/><a NAME="E10:6_3_1_2"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">X1</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="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> or <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/>

<b>assume </b><a NAME="E1:6_3_1_2_1"/>
<font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/>

<a NAME="E2:6_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">XX</font>
 
<b>by </b><i><a class="txt" href="bvfunc_2.html#E10:6"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:6_3_1_2_1"/>
<font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">e</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E9:6_3_1_2"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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="E11:6_3_1_2"/><b>then </b><i><font color="Green" title="E20">A20</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E6:6_3_1_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E12:6_3_1_2"/>
<font color="Maroon" title="c10">e</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font>
 ;<br/><a NAME="E13:6_3_1_2"/><b>then </b>
<font color="Maroon" title="c10">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:6_3"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="bvfunc_2.html#E11:6_3_1_2"><i><font color="Green" title="E20">A20</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><a NAME="E14:6_3_1_2"/><b>then </b><i><font color="Green" title="E21">A21</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">Ad</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E4:6_3"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">ad</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:6_3_1_2"/><i><font color="Green" title="E22">A22</font></i>: 
<font color="Maroon" title="c11">ad</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ad</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E4:6_3"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E17:6_3_1_2"/><i><font color="Green" title="E23">A23</font></i>: 
<font color="Maroon" title="c11">ad</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="bvfunc_2.html#E5:6"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="bvfunc_2.html#E4:6_3"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="bvfunc_2.html#E16:6_3_1_2"><i><font color="Green" title="E22">A22</font></i></a></i>;<br/><a NAME="E18:6_3_1_2"/><i><font color="Green" title="E24">A24</font></i>: 
<font color="Maroon" title="c11">ad</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E14:6_3_1_2"><i><font color="Green" title="E21">A21</font></i></a>, <a class="txt" href="bvfunc_2.html#E16:6_3_1_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">add</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:6_3_1_2"/><i><font color="Green" title="E25">A25</font></i>: 
<font color="Maroon" title="c12">add</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ad</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E17:6_3_1_2"><i><font color="Green" title="E23">A23</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>thus </b><a NAME="E21:6_3_1_2"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font>
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E18:6_3_1_2"><i><font color="Green" title="E24">A24</font></i></a>, <a class="txt" href="bvfunc_2.html#E20:6_3_1_2"><i><font color="Green" title="E25">A25</font></i></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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E16:6"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> &amp;  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">XX</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E10:6"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="bvfunc_2.html#E12:6"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="bvfunc_2.html#E14:6"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="bvfunc_2.html#D2" target="_self" title="BVFUNC_2:def.2">Def2</a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1: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>
