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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">PA</font>, <font color="Olive" title="b2">PB</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">PA</font> <a href="partit1.html#NR2" title="PARTIT1:NR.2">'&gt;'</a> <font color="Olive" title="b2">PB</font> holds <br/>for <font color="Olive" title="b3">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">PA</font> holds <br/><font color="Olive" title="b3">b</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Olive" title="b2">PB</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">PA</font>, <font color="Maroon" title="c3">PB</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">PA</font> <a href="partit1.html#NR2" title="PARTIT1:NR.2">'&gt;'</a> <font color="Maroon" title="c3">PB</font> implies for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</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">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><br/>



<span class="kw">assume </span><a NAME="E1:11"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c2">PA</font> <a href="partit1.html#NR2" title="PARTIT1:NR.2">'&gt;'</a> <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"> for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</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">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><br/>

<a NAME="E2:11"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">PB</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="E3:11"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c2">PA</font> <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a> <font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="partit1.html#T7" target="_self" title="PARTIT1:th.7">Th7</a></span>;<br/>
<a NAME="E4:11"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</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">b</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c3">PB</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">b</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="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> implies <font color="Maroon" title="c4">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><br/>

<span class="kw">assume </span><a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</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">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><br/>

<span class="kw">set </span><font color="Maroon" title="c5">B0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">xb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:11_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
( <font color="Maroon" title="c6">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c6">xb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="setfam_1.html#D3" title="SETFAM_1:def.3">SETFAM_1:def 3</a></span>;<br/>
<a NAME="E4:11_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c6">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/>
<a NAME="E5:11_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x8</font> where <font color="Olive" title="b2">x8</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="Olive" title="b2">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span>  implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 ; <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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font></span><br/>

<a NAME="E2:11_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
 ex <font color="Olive" title="b1">x8</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">x8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> &amp; <font color="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:11_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1_1"><span class="lab"><font color="Green" title="E18">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>
<a NAME="E6:11_1"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E7:11_1"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x8</font> where <font color="Olive" title="b2">x8</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="Olive" title="b2">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_2"/><span class="lab"><font color="Green" title="E19">A13</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:11_1_2"/><span class="lab"><font color="Green" title="E20">A14</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">x1</font>
 <span class="kw">and </span><br/><a NAME="E4:11_1_2"/><span class="lab"><font color="Green" title="E21">A15</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E1:11_1_2"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:11_1_2"/><span class="lab"><font color="Green" title="E22">A16</font></span>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 <span class="kw">and </span><br/><a NAME="E7:11_1_2"/><span class="lab"><font color="Green" title="E23">A17</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E4:11_1_2"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></span>;<br/>
<a NAME="E8:11_1_2"/><span class="lab"><font color="Green" title="E24">A18</font></span>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y1</font> or <font color="Maroon" title="c4">b</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:11_1_2"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<a NAME="E9:11_1_2"/><span class="lab"><font color="Green" title="E25">A19</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_2"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="txt" href="#E3:11_1_2"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="txt" href="#E4:11_1_2"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="txt" href="#E7:11_1_2"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="txt" href="#E8:11_1_2"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E10:11_1_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_2"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="txt" href="#E9:11_1_2"><span class="lab"><font color="Green" title="E25">A19</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="E8:11_1"/><span class="lab"><font color="Green" title="E19">A20</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11_1"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E9:11_1"/><span class="lab"><font color="Green" title="E20">A21</font></span>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x8</font> where <font color="Olive" title="b2">x8</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="Olive" title="b2">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span>  implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_3"/><span class="lab"><font color="Green" title="E21">A22</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 ; <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="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:11_1_3"/><span class="lab"><font color="Green" title="E22">A23</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">and </span><br/><a NAME="E4:11_1_3"/><span class="lab"><font color="Green" title="E23">A24</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_3"><span class="lab"><font color="Green" title="E21">A22</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:11_1_3"/><span class="lab"><font color="Green" title="E24">A25</font></span>: 
 ex <font color="Olive" title="b1">x8</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">x8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_3"><span class="lab"><font color="Green" title="E23">A24</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:11_1_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_3"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="txt" href="#E5:11_1_3"><span class="lab"><font color="Green" title="E24">A25</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="E10:11_1"/><span class="lab"><font color="Green" title="E21">A26</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11_1"><span class="lab"><font color="Green" title="E20">A21</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E11:11_1"/><span class="lab"><font color="Green" title="E22">A27</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x8</font> where <font color="Olive" title="b1">x8</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="Olive" title="b1">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b1">x8</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="txt" href="#E10:11_1"><span class="lab"><font color="Green" title="E21">A26</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E12:11_1"/>
<font color="Maroon" title="c4">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="#E4:11_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="txt" href="#E11:11_1"><span class="lab"><font color="Green" title="E22">A27</font></span></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</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="E5:11"/>
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</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">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="#E4:11"><span class="lab"><font color="Green" title="E12">A4</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>
