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

<b>set </b><font color="Maroon" title="c5">X</font> = <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>;<br/>
<a NAME="E1:44_1_1"/>
<span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> or  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> ) )</span><br/>

<b>assume </b><a NAME="E1:44_1_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:44_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D2</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c11">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:44_1_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D2</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E1:44_1_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c10">y1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="txt" href="#E3:44_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E5:44_1_1_1"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x2</font> = <font color="Maroon" title="c9">x2</font>, <font color="Maroon" title="c15">y2</font> = <font color="Maroon" title="c11">y2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font> <b>by </b><i><a class="txt" href="#E3:44_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E5:44_1_1_1"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">xy1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font><b> such that </b><br/><a NAME="E9:44_1_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c16">xy1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c12">x1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c16">xy1</font> &amp; <font color="Maroon" title="c13">y1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c16">xy1</font> )
 <b>by </b><i><a class="txt" href="#E3:44_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E5:44_1_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">xy2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font><b> such that </b><br/><a NAME="E11:44_1_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon" title="c17">xy2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D2</font> &amp; <font color="Maroon" title="c14">x2</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c17">xy2</font> &amp; <font color="Maroon" title="c15">y2</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c17">xy2</font> )
 <b>by </b><i><a class="txt" href="#E3:44_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E5:44_1_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">S1'</font> = <font color="Maroon" title="c1">S1</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="#E3:44_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">S2'</font> = <font color="Maroon" title="c2">S2</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="#E3:44_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">xy1'</font> = <font color="Maroon" title="c16">xy1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c18">S1'</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c21">xy2'</font> = <font color="Maroon" title="c17">xy2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c19">S2'</font> ;<br/>
<a NAME="E16:44_1_1_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c20">xy1'</font>,<font color="Maroon" title="c21">xy2'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c18">S1'</font>,<font color="Maroon" title="c19">S2'</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">z</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">xy1</font>,<font color="Maroon" title="c17">xy2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> ;<br/>
<b>take </b>
<font color="Maroon" title="c22">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> &amp; <font color="Maroon" title="c22">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c22">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<b>thus </b><a NAME="E18:44_1_1_1"/>
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="#E9:44_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E11:44_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c22">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<a NAME="E19:44_1_1_1"/>
( not <font color="Maroon" title="c1">S1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c2">S2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 
<b>by </b><i><a class="txt" href="#E3:44_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:44_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c22">z</font> &amp; <font color="Maroon" title="c7">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c22">z</font> )
 <b>by </b><i><a class="txt" href="#E3:44_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E5:44_1_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E9:44_1_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E11:44_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:44_1_1"/>
<span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> is  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
