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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1"/>
(  not <font color="Maroon" title="c1">I</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c1">I</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1"/><i><font color="Green" title="E2">A1</font></i>: 
not <font color="Maroon" title="c1">I</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:8_1_1_1_1"/><i><font color="Green" title="E3">A2</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i><b>take </b>
<font color="Maroon" title="c3">X</font>
;<br/><b>thus </b><a NAME="E4:8_1_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff <font color="Olive" title="b1">x</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:8_1_1_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c4">x</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="closure2.html#E3:8_1_1_1_1"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>

<b>assume </b><a NAME="E2:8_1_1_1_1_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c4">x</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c5">xx</font> = <font color="Maroon" title="c4">x</font> as    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="closure2.html#E2:8_1_1_1_1_1"><i><font color="Green" title="E4">A3</font></i></a></i>;<br/><a NAME="E2:8_1_1_1_1_1_1"/><i><font color="Green" title="E5">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:8_1_1_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">xx</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"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="closure2.html#E1:8_1_1_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="closure2.html#T3" target="_self" title="CLOSURE2:th.3">Th3</a></i>;<br/><b>hence </b><a NAME="E4:8_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="closure2.html#E2:8_1_1_1_1_1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><b>thus </b><a NAME="E5:8_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>]
 <b>by </b><i><a class="txt" href="closure2.html#E2:8_1_1_1_1_1"><i><font color="Green" title="E4">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:8_1_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="closure2.html#E3:8_1_1_1_1"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2"/><i><font color="Green" title="E2">A5</font></i>: 
<font color="Maroon" title="c1">I</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_2"/><i><font color="Green" title="E3">A6</font></i>: 
<font color="Maroon" title="c2">M</font> is <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a><br/>

<b>assume </b><a NAME="E1:8_1_1_1_2_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<b>hence </b><a NAME="E2:8_1_1_1_2_1"/>
<font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="closure2.html#E1:8_1_1_1_2"><i><font color="Green" title="E2">A5</font></i></a></i>;<br/>


</div><b>end;</b></div><b>take </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
;<br/><b>thus </b><a NAME="E3:8_1_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> iff <font color="Olive" title="b1">x</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:8_1_1_1_2_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c3">x</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1_1_2_2_1"/><i><font color="Green" title="E4">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">M'</font> = <font color="Maroon" title="c2">M</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <b>by </b><i><a class="txt" href="closure2.html#E1:8_1_1_1_2"><i><font color="Green" title="E2">A5</font></i></a></i>;<br/>
<a NAME="E3:8_1_1_1_2_2_1"/>
 <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c4">M'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:8_1_1_1_2_2_1_1"/>
 <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">M'</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:8_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>hence </b><a NAME="E2:8_1_1_1_2_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">M'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">y</font> =  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c4">M'</font> ;<br/>
<a NAME="E5:8_1_1_1_2_2_1"/>
<font color="Maroon" title="c5">y</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:8_1_1_1_2"><i><font color="Green" title="E2">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:8_1_1_1_2_2_1"/>
<font color="Maroon" title="c3">x</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:8_1_1_1_2_2_1"><i><font color="Green" title="E4">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:8_1_1_1_2_2"/>
<font color="Maroon" title="c3">x</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E4:8_1_1_1_2_2"/><i><font color="Green" title="E4">A8</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 ;<br/>
<a NAME="E5:8_1_1_1_2_2"/>
<font color="Maroon" title="c4">y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E6:8_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c4">y</font> is <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>
 
<b>by </b><i><a class="txt" href="closure2.html#E2:8_1_1_1_2"><i><font color="Green" title="E3">A6</font></i></a>, <a class="ref" href="pboole.html#T142" title="PBOOLE:th.142">PBOOLE:142</a></i>;<br/>
<a NAME="E7:8_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="closure2.html#E1:8_1_1_1_2"><i><font color="Green" title="E2">A5</font></i></a>, <a class="ref" href="pboole.html#T141" title="PBOOLE:th.141">PBOOLE:141</a></i>;<br/>
<b>hence </b><a NAME="E8:8_1_1_1_2_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="closure2.html#E4:8_1_1_1_2_2"><i><font color="Green" title="E4">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
