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

<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>;<br/>



<b>set </b><font color="Maroon">D1</font> =  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">D2</font> =  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font>;<br/>
<a NAME="E1:46"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:46"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font></span>)</span></span><a href="yellow_3.html#K6" target="_self">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:46"/>
 <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> is <a href="waybel_0.html#V1">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">D1'</font> =  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font> ;<br/>
<b>reconsider </b><font color="Maroon">D2'</font> =  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S2</font> <b>by </b><i>, <a class="ref" href="funct_5.html#T19">FUNCT_5:19</a></i>;<br/>
<b>consider </b><font color="Maroon">q1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:46_1"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">q1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q1</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D2'</font> <b>by </b><i>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">q2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:46_1"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">q2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">q2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D2'</font> <b>by </b><i>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span><b> such that </b><br/><a NAME="E11:46_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">q1</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">q2</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">zz</font> = <font color="Maroon">z</font> <a href="yellow_3.html#K8" target="_self">`1</a>  as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D1'</font> <b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z</font> <a href="yellow_3.html#K9" target="_self">`2</a>  as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D2'</font> <b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>take </b>
<font color="Maroon">zz</font>
;<br/>

<b>thus </b><a NAME="E14:46_1"/>
<font color="Maroon">zz</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E15:46_1"/>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1'</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E16:46_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">zz</font>,<font color="Maroon">z2</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T8">MCART_1:8</a></i>;<br/>
<b>hence </b><a NAME="E17:46_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">zz</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">zz</font> )
 <b>by </b><i>, </i>;<br/>


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

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

<b>assume </b><a NAME="E4:46"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">D1'</font> =  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font> <b>by </b><i><a class="ref" href="funct_5.html#T19">FUNCT_5:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D2'</font> =  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S2</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">q1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:46"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q1</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D1'</font> <b>by </b><i>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">q2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:46"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">q2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D1'</font> <b>by </b><i>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span><b> such that </b><br/><a NAME="E14:46"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">zz</font> = <font color="Maroon">z</font> <a href="yellow_3.html#K9" target="_self">`2</a>  as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D2'</font> <b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z</font> <a href="yellow_3.html#K8" target="_self">`1</a>  as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D1'</font> <b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>take </b>
<font color="Maroon">zz</font>
;<br/>

<b>thus </b><a NAME="E17:46"/>
<font color="Maroon">zz</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E18:46"/>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1'</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E19:46"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">zz</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T8">MCART_1:8</a></i>;<br/>
<b>hence </b><a NAME="E20:46"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">zz</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">zz</font> )
 <b>by </b><i>, </i>;<br/>


</div>
