<?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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span><br/>  for <font color="Olive" title="b2">PA</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">PA</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b2">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Olive" title="b1">G</font></span><br/><span class="kw">let </span><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="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">PA</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Olive" title="b1">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">PA</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">PA</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">G</font> implies <font color="Maroon" title="c3">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:9"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c3">PA</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font></span><br/>

<a NAME="E2:9"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">PA</font> holds <br/> ex <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">b</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">PA</font> implies  ex <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">a</font>;<br/>
<a NAME="E1:9_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font></span>)</span> <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#D4" onmouseover="rs('eqrel_1/D4')" onmouseout="rh()">EQREL_1:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:9_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">PA</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> )</span><br/>

<a NAME="E3:9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c4">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="ref" href="eqrel_1.html#D4" onmouseover="rs('eqrel_1/D4')" onmouseout="rh()">EQREL_1:def 4</a></span>;<br/>
<a NAME="E4:9_1"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E6:9_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">and </span><br/><a NAME="E7:9_1"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E8:9_1"/>
<font color="Maroon" title="c6">b</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E7:9_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="bvfunc_2.html#D3" target="_self" onmouseover="rs('bvfunc_2/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E10:9_1"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">PA</font>
 <span class="kw">and </span><br/><a NAME="E11:9_1"/>
<font color="Maroon" title="c7">B</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="E12:9_1"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<font color="Maroon" title="c6">b</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="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="partit1.html#D1" onmouseover="rs('partit1/D1')" onmouseout="rh()">PARTIT1:def 1</a></span>;<br/>
<a NAME="E13:9_1"/>
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c8">u</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:9_1_1"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">u</font>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E12:9_1"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E4:9_1_1"/><span class="lab"><font color="Green" title="E15">A11</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">u</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="#E3:9_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E2:9_1_1"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:9_1_1"/>
<font color="Maroon" title="c8">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">PA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E3:9_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:9_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E3:9_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E4:9_1_1"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="eqrel_1.html#D4" onmouseover="rs('eqrel_1/D4')" onmouseout="rh()">EQREL_1:def 4</a>, <a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:9_1"/>
 ex <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E12:9_1"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:9"/>
<font color="Maroon" title="c3">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a>  <a href="bvfunc_2.html#K3" title="BVFUNC_2:func.3">'\/'</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D2" onmouseover="rs('setfam_1/D2')" onmouseout="rh()">SETFAM_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
