<?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#V4">antisymmetric</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#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">D</font> be   <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/>



<a NAME="E1:75"/><i><font color="Green">E18</font></i>: 
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="E2:75"/><b>then </b><i><font color="Green">E20</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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:75_1"/><i><font color="Green">E25</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font>,<font color="Maroon">S1</font>
 <b>and </b><br/><a NAME="E2:75_1"/><i><font color="Green">E28</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font>,<font color="Maroon">S2</font>
 ;<br/><a NAME="E3:75_1"/>
 ex <font color="Olive">a</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> st <br/>( <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">a</font> &amp; ( for <font color="Olive">b</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>  st <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font><b> such that </b><br/><a NAME="E2:75_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E3:75_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font>  st  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">x1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/>
<b>consider </b><font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font><b> such that </b><br/><a NAME="E5:75_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E6:75_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font>  st  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">x2</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E1:75_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/>
<b>take </b><font color="Maroon">a</font> = <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>;<br/>

<b>thus </b><a NAME="E7:75_1_1"/>
<font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:75_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">q2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:75_1_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">q1</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">q2</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">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</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/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font> <b>by </b><i><a class="txt" href="yellow_3.html#E3:75_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">q2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font> <b>by </b><i><a class="txt" href="yellow_3.html#E3:75_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E6:75_1_1_1"/>
( <font color="Maroon">q1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">q2</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, <a class="txt" href="yellow_3.html#E2:75_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_3.html#E3:75_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:75_1_1_1"/>
<font color="Maroon">q</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E3:75_1_1"><i><font color="Green">E30</font></i></a>, </i>;<br/>


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

<b>let </b><font color="Maroon">b</font> be   <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>;<br/>

<b>assume </b><a NAME="E8:75_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E9:75_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E10:75_1_1"/><b>then </b>
(  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font> <a href="yellow_3.html#K8" target="_self">`1</a>  &amp;  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font> <a href="yellow_3.html#K9" target="_self">`2</a>  )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E3:75_1_1"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E11:75_1_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> <a href="yellow_3.html#K8" target="_self">`1</a>  &amp; <font color="Maroon">x2</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> <a href="yellow_3.html#K9" target="_self">`2</a>  )
 
<b>by </b><i>, <a class="txt" href="yellow_3.html#E2:75_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:75_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:75_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">D</font>,<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>by </b><i><a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:75"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">D</font>,<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>then consider </b><font color="Maroon">x</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="E6:75"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E7:75"/><i><font color="Green">E116</font></i>: 
for <font color="Olive">b</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>  st <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/>
<a NAME="E8:75"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E9:75"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:75"/><i><font color="Green">E119</font></i>: 
 <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:75"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font>  st  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font>;<br/>

<b>assume </b><a NAME="E1:75_2"/>
 <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:75_2"/><b>then </b>
<font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:75_2"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:75_2"/>
<font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:75"/>
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">D</font>,<font color="Maroon">S1</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/>

<a NAME="E13:75"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font>  st  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font>;<br/>

<b>assume </b><a NAME="E1:75_3"/>
 <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:75_3"/><b>then </b>
<font color="Maroon">D</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:75_3"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:75_3"/>
<font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:75"/>
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">D</font>,<font color="Maroon">S2</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/>


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