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

<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  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> ;<br/><b>let </b><font color="Maroon" title="c3">D1</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c4">D2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E8">A1</font></i>: 
<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#V1" title="WAYBEL_0:attr.1">directed</a>
 ;<br/>

<b>thus </b><a NAME="E2:43"/>
<font color="Maroon" title="c3">D1</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">q1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">D2</font>;<br/>
<a NAME="E3:43_1"/>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">q1</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <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; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">q1</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <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="yellow_3.html#E1:43_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">z</font> being   <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><b> such that </b><br/><a NAME="E5:43_1"/><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c8">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; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">q1</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">q1</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="txt" href="yellow_3.html#E1:43"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">zz</font> = <font color="Maroon" title="c8">z</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">D1</font> <b>by </b><i><a class="txt" href="yellow_3.html#E5:43_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">z2</font> = <font color="Maroon" title="c8">z</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">D2</font> <b>by </b><i><a class="txt" href="yellow_3.html#E5:43_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">zz</font>
;<br/>

<b>thus </b><a NAME="E8:43_1"/>
<font color="Maroon" title="c9">zz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font>
 ;<br/>

<a NAME="E9:43_1"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D2</font> &amp; <font color="Maroon" title="c8">z</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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E5:43_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E10:43_1"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">q1</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c9">zz</font>,<font color="Maroon" title="c10">z2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">q1</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c9">zz</font>,<font color="Maroon" title="c10">z2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E5:43_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></i>;<br/>
<b>hence </b><a NAME="E11:43_1"/>
( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">zz</font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">zz</font> )
 <b>by </b><i><a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E3:43"/><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c5">x</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">q1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">D1</font>;<br/>
<a NAME="E5:43"/>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <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; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <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="yellow_3.html#E3:43"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">z</font> being   <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><b> such that </b><br/><a NAME="E7:43"/><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c8">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; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="txt" href="yellow_3.html#E1:43"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">zz</font> = <font color="Maroon" title="c8">z</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">D2</font> <b>by </b><i><a class="txt" href="yellow_3.html#E7:43"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">z2</font> = <font color="Maroon" title="c8">z</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">D1</font> <b>by </b><i><a class="txt" href="yellow_3.html#E7:43"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">zz</font>
;<br/>

<b>thus </b><a NAME="E10:43"/>
<font color="Maroon" title="c9">zz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D2</font>
 ;<br/>

<a NAME="E11:43"/>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D2</font> &amp; <font color="Maroon" title="c8">z</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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E7:43"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E12:43"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c5">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">z2</font>,<font color="Maroon" title="c9">zz</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c6">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">z2</font>,<font color="Maroon" title="c9">zz</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E7:43"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></i>;<br/>
<b>hence </b><a NAME="E13:43"/>
( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">zz</font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">zz</font> )
 <b>by </b><i><a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></i>;<br/>


</div>
