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

<b>let </b><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c3">Con</font> be    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">R</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">R</font>;<br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c1">P</font>;<br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">Con</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span>
 ;<br/>

<div><i><font color="Green" title="E17">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon" title="c3">Con</font> is <a href="conlat_1.html#V5" title="CONLAT_1:attr.5">co-Galois</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">f'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">R</font>, <font color="Maroon" title="c7">g'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c3">Con</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">f'</font>,<font color="Maroon" title="c7">g'</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> &amp; <font color="Maroon" title="c6">f'</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> &amp; <font color="Maroon" title="c7">g'</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> &amp; ( for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><br/> for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font> holds <br/> ( <font color="Olive" title="b1">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">g'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p1</font></span>)</span> &amp; <font color="Olive" title="b3">r1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">r1</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="conlat_1.html#D10" target="_self" title="CONLAT_1:def.10">Def10</a></i>;<br/><i><font color="Green" title="E19">A4</font></i>: <font color="Maroon" title="c4">f</font> = 
<font color="Maroon" title="c3">Con</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:29"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c6">f'</font>
<b>by </b><i><a class="txt" href="conlat_1.html#E3:29_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/><i><font color="Green" title="E20">A5</font></i>: <font color="Maroon" title="c5">g</font> = 
<font color="Maroon" title="c3">Con</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:29"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c7">g'</font>
<b>by </b><i><a class="txt" href="conlat_1.html#E3:29_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/><a NAME="E6:29_1"/><i><font color="Green" title="E21">A6</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>  st <font color="Olive" title="b1">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b2">r</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>;<br/><b>let </b><font color="Maroon" title="c9">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>;<br/>



<b>assume </b><a NAME="E1:29_1_3"/>
<font color="Maroon" title="c8">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">r</font>
 ;<br/>

<a NAME="E2:29_1_3"/><b>then </b><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E3:29_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="conlat_1.html#E4:29_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="waybel_0.html#D5" title="WAYBEL_0:def.5">WAYBEL_0:def 5</a></i>;<br/>
<a NAME="E3:29_1_3"/>
<font color="Maroon" title="c9">r</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E3:29_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="conlat_1.html#E4:29_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="conlat_1.html#E5:29_1"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_3"/>
<font color="Maroon" title="c9">r</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E2:29_1_3"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:29_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>  st <font color="Olive" title="b2">r</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Olive" title="b1">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>;<br/><b>let </b><font color="Maroon" title="c9">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>;<br/>



<b>assume </b><a NAME="E1:29_1_4"/>
<font color="Maroon" title="c9">r</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
 ;<br/>

<a NAME="E2:29_1_4"/><b>then </b><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">r</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E3:29_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="conlat_1.html#E5:29_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="waybel_0.html#D5" title="WAYBEL_0:def.5">WAYBEL_0:def 5</a></i>;<br/>
<a NAME="E3:29_1_4"/>
<font color="Maroon" title="c8">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E3:29_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="conlat_1.html#E4:29_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="conlat_1.html#E5:29_1"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_4"/>
<font color="Maroon" title="c8">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">r</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E2:29_1_4"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:29_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font> iff <font color="Olive" title="b2">r</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E6:29_1"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_2"/><i><font color="Green" title="E18">A9</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font> iff <font color="Olive" title="b2">r</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> )
 ;<br/><a NAME="E2:29_2"/><b>then </b><i><font color="Green" title="E19">A10</font></i>: 
for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><br/> for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font> holds <br/> ( <font color="Olive" title="b1">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p1</font></span>)</span> &amp; <font color="Olive" title="b3">r1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">r1</font></span>)</span> )
 ;<br/><a NAME="E3:29_2"/>
for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">p2</font> holds <br/>for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>  st <font color="Olive" title="b3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p2</font> holds <br/><font color="Olive" title="b3">r1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b4">r2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>;<br/>

<b>assume </b><a NAME="E1:29_2_1"/><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c6">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">p2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">r1</font>, <font color="Maroon" title="c9">r2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>;<br/>

<b>assume </b><a NAME="E2:29_2_1"/><i><font color="Green" title="E21">A12</font></i>: 
( <font color="Maroon" title="c8">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p1</font> &amp; <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p2</font> )
 ;<br/>

<a NAME="E3:29_2_1"/>
<font color="Maroon" title="c7">p2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:29_2"><i><font color="Green" title="E18">A9</font></i></a></i>;<br/>
<a NAME="E4:29_2_1"/><b>then </b>
<font color="Maroon" title="c6">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:29_2_1"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E5:29_2_1"/>
<font color="Maroon" title="c8">r1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">r2</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:29_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="conlat_1.html#E2:29_2_1"><i><font color="Green" title="E21">A12</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:29_2"/><b>then </b><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D5" title="WAYBEL_0:def.5">WAYBEL_0:def 5</a></i>;<br/><a NAME="E5:29_2"/>
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>  st <font color="Olive" title="b1">r1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">r2</font> holds <br/>for <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r2</font> holds <br/><font color="Olive" title="b3">p1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b4">p2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">r1</font>, <font color="Maroon" title="c7">r2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font>;<br/>

<b>assume </b><a NAME="E1:29_2_2"/><i><font color="Green" title="E21">A14</font></i>: 
<font color="Maroon" title="c6">r1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">r2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>;<br/>

<b>assume </b><a NAME="E2:29_2_2"/><i><font color="Green" title="E22">A15</font></i>: 
( <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">r1</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">r2</font> )
 ;<br/>

<a NAME="E3:29_2_2"/>
<font color="Maroon" title="c7">r2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:29_2"><i><font color="Green" title="E18">A9</font></i></a></i>;<br/>
<a NAME="E4:29_2_2"/><b>then </b>
<font color="Maroon" title="c6">r1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:29_2_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E5:29_2_2"/>
<font color="Maroon" title="c8">p1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">p2</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:29_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="conlat_1.html#E2:29_2_2"><i><font color="Green" title="E22">A15</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:29_2"/><b>then </b>
<font color="Maroon" title="c5">g</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D5" title="WAYBEL_0:def.5">WAYBEL_0:def 5</a></i>;<br/><b>hence </b><a NAME="E7:29_2"/>
<font color="Maroon" title="c3">Con</font> is <a href="conlat_1.html#V5" title="CONLAT_1:attr.5">co-Galois</a>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:29"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="conlat_1.html#E2:29_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="conlat_1.html#E4:29_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="conlat_1.html#D10" target="_self" title="CONLAT_1:def.10">Def10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:29"/>
( <font color="Maroon" title="c3">Con</font> is <a href="conlat_1.html#V5" title="CONLAT_1:attr.5">co-Galois</a> iff for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font> iff <font color="Olive" title="b2">r</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> ) )
 <b>by </b><i><a class="txt" href="conlat_1.html#E2:29"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>


</div>
