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

<a NAME="E1:24_1_1"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<a NAME="E2:24_1_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
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="c2">PA</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="c2">PA</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> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:24_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E5:24_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff <font color="Maroon" title="c5">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font> )</span><br/>

<a NAME="E1:24_1_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font> holds <br/><font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font> implies <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c6">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font></span><br/>

<a NAME="E2:24_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c6">d</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/>
<a NAME="E3:24_1_1_1_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 ex <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font> &amp; <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; <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></span>;<br/>
<a NAME="E4:24_1_1_1_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon" title="c6">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">PA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:24_1_1_1_1"/>
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E4:24_1_1_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:24_1_1_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
( <font color="Maroon" title="c5">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font> implies ( <font color="Maroon" title="c5">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:24_1_1_1"/>
( <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff <font color="Maroon" title="c5">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E2:24_1_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">take </span>
<font color="Maroon" title="c4">X</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font></span>)</span> &amp; <font color="Maroon" title="c4">X</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font> ) ) )</span><br/>

<a NAME="E6:24_1_1"/><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1_2"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">X</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">a</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E3:24_1_1_2"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E4:24_1_1_2"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_2"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="partit1.html#T13" target="_self" title="PARTIT1:th.13">Th13</a></span>;<br/>
<a NAME="E5:24_1_1_2"/><span class="lab"><font color="Green" title="E24">A15</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E4:24_1_1_2"><span class="lab"><font color="Green" title="E23">A14</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:24_1_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_2"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E5:24_1_1_2"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:24_1_1"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1_3"/><span class="lab"><font color="Green" title="E22">A17</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:24_1_1_3"/><span class="lab"><font color="Green" title="E23">A18</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E4:24_1_1_3"/><span class="lab"><font color="Green" title="E24">A19</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_3"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:24_1_1_3"/><span class="lab"><font color="Green" title="E25">A20</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E4:24_1_1_3"><span class="lab"><font color="Green" title="E24">A19</font></span></a></span>;<br/>
<a NAME="E6:24_1_1_3"/><span class="lab"><font color="Green" title="E26">A21</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1_3"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/>
<a NAME="E7:24_1_1_3"/><span class="lab"><font color="Green" title="E27">A22</font></span>: 
 ex <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font> &amp; <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; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_1_3"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></span>;<br/>
<a NAME="E8:24_1_1_3"/><span class="lab"><font color="Green" title="E28">A23</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E7:24_1_1_3"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:24_1_1_3"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_3"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="txt" href="#E8:24_1_1_3"><span class="lab"><font color="Green" title="E28">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:24_1_1"/><span class="lab"><font color="Green" title="E22">A24</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E7:24_1_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E9:24_1_1"/><span class="lab"><font color="Green" title="E23">A25</font></span>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  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="c4">X</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> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies ( <font color="Maroon" title="c5">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="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> or <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">B</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1_4"/><span class="lab"><font color="Green" title="E24">A26</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> or <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">B</font> ) ) )</span><br/>

<a NAME="E2:24_1_1_4"/><span class="lab"><font color="Green" title="E25">A27</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E1:24_1_1_4"><span class="lab"><font color="Green" title="E24">A26</font></span></a></span>;<br/>
<a NAME="E3:24_1_1_4"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_4"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/>
<a NAME="E4:24_1_1_4"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_4"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">Aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:24_1_1_4"/><span class="lab"><font color="Green" title="E28">A30</font></span>: 
<font color="Maroon" title="c6">Aa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font>
 <span class="kw">and </span><br/><a NAME="E7:24_1_1_4"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
<font color="Maroon" title="c6">Aa</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> 
 <span class="kw">and </span><br/><a NAME="E8:24_1_1_4"/><span class="lab"><font color="Green" title="E30">A32</font></span>: 
<font color="Maroon" title="c5">A</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="c6">Aa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_4"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">aa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:24_1_1_4"/><span class="lab"><font color="Green" title="E31">A33</font></span>: 
<font color="Maroon" title="c7">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Aa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1_4"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E11:24_1_1_4"/><span class="lab"><font color="Green" title="E32">A34</font></span>: 
<font color="Maroon" title="c7">aa</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E6:24_1_1_4"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E10:24_1_1_4"><span class="lab"><font color="Green" title="E31">A33</font></span></a></span>;<br/>
<a NAME="E12:24_1_1_4"/><span class="lab"><font color="Green" title="E33">A35</font></span>: 
<font color="Maroon" title="c7">aa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">Aa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24_1_1_4"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<span class="kw">thus </span><a NAME="E13:24_1_1_4"/>
<font color="Maroon" title="c5">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1_1_4"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E11:24_1_1_4"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="txt" href="#E12:24_1_1_4"><span class="lab"><font color="Green" title="E33">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> or <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">B</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> or <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E14:24_1_1_4"/><span class="lab"><font color="Green" title="E34">A36</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> or <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<a NAME="E15:24_1_1_4"/><span class="lab"><font color="Green" title="E35">A37</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E14:24_1_1_4"><span class="lab"><font color="Green" title="E34">A36</font></span></a></span>;<br/>
<a NAME="E16:24_1_1_4"/><span class="lab"><font color="Green" title="E36">A38</font></span>: 
( <font color="Maroon" title="c8">B</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c8">B</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c3">PB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24_1_1_4"><span class="lab"><font color="Green" title="E35">A37</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E37">A39</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_1_4_1"/><span class="lab"><font color="Green" title="E38">A40</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font></span><br/><a NAME="E2:24_1_1_4_1"/><span class="lab"><font color="Green" title="E39">A41</font></span>: 
( <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c3">PB</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_4"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E4:24_1_1_4"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E16:24_1_1_4"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="txt" href="#E1:24_1_1_4_1"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="ref" href="partit1.html#T11" target="_self" title="PARTIT1:th.11">Th11</a></span>;<br/><a NAME="E3:24_1_1_4_1"/><span class="lab"><font color="Green" title="E40">A42</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E4:24_1_1_4_1"/><span class="lab"><font color="Green" title="E41">A43</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_4"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:24_1_1_4_1"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="txt" href="#E3:24_1_1_4_1"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/><a NAME="E5:24_1_1_4_1"/><span class="lab"><font color="Green" title="E42">A44</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1_4_1_1"/><span class="lab"><font color="Green" title="E43">A45</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font></span><br/>

<span class="kw">thus </span><a NAME="E2:24_1_1_4_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_1_4_1"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="txt" href="#E1:24_1_1_4_1_1"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:24_1_1_4_1"/><span class="lab"><font color="Green" title="E43">A46</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/><a NAME="E7:24_1_1_4_1"/><span class="lab"><font color="Green" title="E44">A47</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24_1_1_4"><span class="lab"><font color="Green" title="E35">A37</font></span></a>, <a class="txt" href="#E2:24_1_1_4_1"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="txt" href="#E6:24_1_1_4_1"><span class="lab"><font color="Green" title="E43">A46</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/><a NAME="E8:24_1_1_4_1"/><span class="lab"><font color="Green" title="E45">A48</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_4_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1_4_1_2"/><span class="lab"><font color="Green" title="E46">A49</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font></span><br/>

<span class="kw">thus </span><a NAME="E2:24_1_1_4_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1_4_1"><span class="lab"><font color="Green" title="E44">A47</font></span></a>, <a class="txt" href="#E1:24_1_1_4_1_2"><span class="lab"><font color="Green" title="E46">A49</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E9:24_1_1_4_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1_4_1"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E8:24_1_1_4_1"><span class="lab"><font color="Green" title="E45">A48</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E18:24_1_1_4"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> or <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24_1_1_4"><span class="lab"><font color="Green" title="E37">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:24_1_1"/><span class="lab"><font color="Green" title="E24">A50</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1_5"/><span class="lab"><font color="Green" title="E25">A51</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font></span><br/>

<a NAME="E2:24_1_1_5"/><span class="lab"><font color="Green" title="E26">A52</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E1:24_1_1_5"><span class="lab"><font color="Green" title="E25">A51</font></span></a></span>;<br/>
<a NAME="E3:24_1_1_5"/><span class="lab"><font color="Green" title="E27">A53</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_5"><span class="lab"><font color="Green" title="E26">A52</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/>
<a NAME="E4:24_1_1_5"/><span class="lab"><font color="Green" title="E28">A54</font></span>: 
 ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font> &amp; <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; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_1_5"><span class="lab"><font color="Green" title="E27">A53</font></span></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></span>;<br/>
<a NAME="E5:24_1_1_5"/><span class="lab"><font color="Green" title="E29">A55</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E4:24_1_1_5"><span class="lab"><font color="Green" title="E28">A54</font></span></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:24_1_1_5"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1_5"><span class="lab"><font color="Green" title="E29">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c4">X</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24_1_1"><span class="lab"><font color="Green" title="E24">A50</font></span></a></span>;<br/>
<a NAME="E12:24_1_1"/><span class="lab"><font color="Green" title="E25">A56</font></span>: 
<font color="Maroon" title="c5">X</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1_1"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E9:24_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<span class="kw">thus </span><a NAME="E13:24_1_1"/>
( <font color="Maroon" title="c4">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font></span>)</span> &amp; <font color="Maroon" title="c4">X</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> &amp; ( for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E12:24_1_1"><span class="lab"><font color="Green" title="E25">A56</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
