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

<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c3">f1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/><b>let </b><font color="Maroon" title="c4">g1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c5">f2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">g2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>;<br/>









<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">g2</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E33">A2</font></i>: 
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">g1</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>
 ;<br/><a NAME="E2:57_1"/><b>then </b>
( <font color="Maroon" title="c4">g1</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c3">f1</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> iff <font color="Maroon" title="c4">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">s</font> ) ) )
 <b>by </b><i><a class="ref" href="waybel_1.html#T9" title="WAYBEL_1:th.9">WAYBEL_1:9</a></i>;<br/><a NAME="E3:57_1"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
( <font color="Maroon" title="c6">g2</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c5">f2</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="yellow_7.html#T42" target="_self" title="YELLOW_7:th.42">Th42</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>;<br/><a NAME="E1:57_1_1"/>
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">s</font></span>)</span></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">t</font></span>)</span></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">t</font></span>)</span> &amp; (  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">s</font></span>)</span> implies <font color="Maroon" title="c4">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">s</font> ) &amp; ( <font color="Maroon" title="c4">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">s</font> implies  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c8">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c7">s</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:57_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="waybel_1.html#T9" title="WAYBEL_1:th.9">WAYBEL_1:9</a></i>;<br/><b>hence </b><a NAME="E2:57_1_1"/>
( <font color="Maroon" title="c6">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">t</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">s</font> iff <font color="Maroon" title="c8">t</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">s</font> )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="yellow_7.html#T2" target="_self" title="YELLOW_7:th.2">Th2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:57_1"/>
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">g2</font>,<font color="Maroon" title="c5">f2</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>
 <b>by </b><i><a class="txt" href="yellow_7.html#E3:57_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="waybel_1.html#T9" title="WAYBEL_1:th.9">WAYBEL_1:9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:57"/><i><font color="Green" title="E33">A4</font></i>: 
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">g2</font>,<font color="Maroon" title="c5">f2</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>
 ;<br/>

<a NAME="E4:57"/><b>then </b>
( <font color="Maroon" title="c6">g2</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c5">f2</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; ( for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span><br/> for <font color="Olive" title="b2">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font> iff <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">t</font> ) ) )
 
<b>by </b><i><a class="ref" href="waybel_1.html#T9" title="WAYBEL_1:th.9">WAYBEL_1:9</a></i>;<br/>
<a NAME="E5:57"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
( <font color="Maroon" title="c4">g1</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c3">f1</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 
<b>by </b><i><a class="txt" href="yellow_7.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="yellow_7.html#T42" target="_self" title="YELLOW_7:th.42">Th42</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>;<br/><b>let </b><font color="Maroon" title="c8">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><a NAME="E1:57_2"/>
( <font color="Maroon" title="c8">s</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c7">t</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> &amp; ( <font color="Maroon" title="c8">s</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> implies <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  ) &amp; ( <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  implies <font color="Maroon" title="c8">s</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> ) )
 <b>by </b><i><a class="txt" href="yellow_7.html#E3:57"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="waybel_1.html#T9" title="WAYBEL_1:th.9">WAYBEL_1:9</a></i>;<br/><b>hence </b><a NAME="E2:57_2"/>
( <font color="Maroon" title="c7">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> iff <font color="Maroon" title="c4">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">s</font> )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="yellow_7.html#T2" target="_self" title="YELLOW_7:th.2">Th2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:57"/>
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">g1</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>
 <b>by </b><i><a class="txt" href="yellow_7.html#E5:57"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="waybel_1.html#T9" title="WAYBEL_1:th.9">WAYBEL_1:9</a></i>;<br/>


</div>
