<?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#V5" title="ORDERS_2:attr.5">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">D</font> be   <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>;<br/>



<a NAME="E1:74"/><i><font color="Green" title="E22">A1</font></i>: 
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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" target="_self" title="YELLOW_3:def.2">Def2</a></i>;<br/>
<a NAME="E2:74"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font></span>)</span></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T1" target="_self" title="YELLOW_3:th.1">Th1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:74_1"/><i><font color="Green" title="E24">A3</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c1">S1</font>
 <b>and </b><br/><a NAME="E2:74_1"/><i><font color="Green" title="E25">A4</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c2">S2</font>
 ;<br/><a NAME="E3:74_1"/>
 ex <font color="Olive" title="b1">a</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> st <br/>( <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">a</font> &amp; ( for <font color="Olive" title="b2">b</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>  st <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">b</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c4">x1</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="E2:74_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
 <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c4">x1</font>
 <b>and </b><br/><a NAME="E3:74_1_1"/><i><font color="Green" title="E27">A6</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>  st  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c4">x1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E1:74_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="yellow_0.html#T16" title="YELLOW_0:th.16">YELLOW_0:16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x2</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="E5:74_1_1"/><i><font color="Green" title="E28">A7</font></i>: 
 <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c5">x2</font>
 <b>and </b><br/><a NAME="E6:74_1_1"/><i><font color="Green" title="E29">A8</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>  st  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c5">x2</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E2:74_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="yellow_0.html#T16" title="YELLOW_0:th.16">YELLOW_0:16</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">a</font> = <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>;<br/>

<b>thus </b><a NAME="E7:74_1_1"/>
<font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">q</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="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:74_1_1_1"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">q1</font>, <font color="Maroon" title="c9">q2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:74_1_1_1"/><i><font color="Green" title="E30">A9</font></i>: 
( <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c9">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c7">q</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">q1</font>,<font color="Maroon" title="c9">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="yellow_3.html#E2:74"><i><font color="Green" title="E23">A2</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="c10">q1</font> = <font color="Maroon" title="c8">q1</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="yellow_3.html#E3:74_1_1_1"><i><font color="Green" title="E30">A9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">q2</font> = <font color="Maroon" title="c9">q2</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="yellow_3.html#E3:74_1_1_1"><i><font color="Green" title="E30">A9</font></i></a></i>;<br/>
<a NAME="E6:74_1_1_1"/>
( <font color="Maroon" title="c10">q1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c11">q2</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">x2</font> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E2:74_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="yellow_3.html#E5:74_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="yellow_3.html#E3:74_1_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:74_1_1_1"/>
<font color="Maroon" title="c7">q</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E3:74_1_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <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="c7">b</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>;<br/>

<b>assume </b><a NAME="E8:74_1_1"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c7">b</font>
 ;<br/>

<a NAME="E9:74_1_1"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></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#E1:74"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E10:74_1_1"/><b>then </b>
(  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c7">b</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  &amp;  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c7">b</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E8:74_1_1"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="yellow_3.html#T34" target="_self" title="YELLOW_3:th.34">Th34</a></i>;<br/>
<a NAME="E11:74_1_1"/><b>then </b>
( <font color="Maroon" title="c4">x1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  &amp; <font color="Maroon" title="c5">x2</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E3:74_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="yellow_3.html#E6:74_1_1"><i><font color="Green" title="E29">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:74_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E9:74_1_1"><i><font color="Green" title="E31">A11</font></i></a>, <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>hence </b><a NAME="E4:74_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">D</font>,<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>by </b><i><a class="ref" href="yellow_0.html#T16" title="YELLOW_0:th.16">YELLOW_0:16</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:74"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">D</font>,<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>then consider </b><font color="Maroon" title="c4">x</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="E6:74"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c4">x</font>
 <b>and </b><br/><a NAME="E7:74"/><i><font color="Green" title="E25">A13</font></i>: 
for <font color="Olive" title="b1">b</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>  st <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T16" title="YELLOW_0:th.16">YELLOW_0:16</a></i>;<br/>
<a NAME="E8:74"/><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></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#E1:74"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E9:74"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
 <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
 
<b>by </b><i><a class="txt" href="yellow_3.html#E6:74"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="yellow_3.html#T34" target="_self" title="YELLOW_3:th.34">Th34</a></i>;<br/>
<a NAME="E10:74"/><i><font color="Green" title="E28">A16</font></i>: 
 <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
 
<b>by </b><i><a class="txt" href="yellow_3.html#E6:74"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="yellow_3.html#E8:74"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="yellow_3.html#T34" target="_self" title="YELLOW_3:th.34">Th34</a></i>;<br/>
<a NAME="E11:74"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>  st  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>;<br/>

<b>assume </b><a NAME="E1:74_2"/>
 <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c5">b</font>
 ;<br/>

<a NAME="E2:74_2"/><b>then </b>
<font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></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#E10:74"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="yellow_3.html#T34" target="_self" title="YELLOW_3:th.34">Th34</a></i>;<br/>
<a NAME="E3:74_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></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#E7:74"><i><font color="Green" title="E25">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:74_2"/>
<font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E8:74"><i><font color="Green" title="E26">A14</font></i></a>, <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>hence </b><a NAME="E12:74"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c1">S1</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E9:74"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="yellow_0.html#T16" title="YELLOW_0:th.16">YELLOW_0:16</a></i>;<br/>

<a NAME="E13:74"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>  st  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>;<br/>

<b>assume </b><a NAME="E1:74_3"/>
 <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c5">b</font>
 ;<br/>

<a NAME="E2:74_3"/><b>then </b>
<font color="Maroon" title="c3">D</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<font color="Maroon" title="c5">b</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#E9:74"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="yellow_3.html#T34" target="_self" title="YELLOW_3:th.34">Th34</a></i>;<br/>
<a NAME="E3:74_3"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<font color="Maroon" title="c5">b</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#E7:74"><i><font color="Green" title="E25">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:74_3"/>
<font color="Maroon" title="c4">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E8:74"><i><font color="Green" title="E26">A14</font></i></a>, <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>hence </b><a NAME="E14:74"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c2">S2</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E10:74"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="yellow_0.html#T16" title="YELLOW_0:th.16">YELLOW_0:16</a></i>;<br/>


</div>
