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<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lattice3.html#K7">~</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="lattice3.html#K7">~</a> </span>)</span>;<br/><b>let </b><font color="Maroon">g2</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="lattice3.html#K7">~</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lattice3.html#K7">~</a> </span>)</span>;<br/>









<b>assume </b><a NAME="E1:57"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">f1</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">g1</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">g2</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">g1</font></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 ;<br/><a NAME="E2:57_1"/><b>then </b>
( <font color="Maroon">g1</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">f1</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><br/> for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> iff <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s</font> ) ) )
 <b>by </b><i><a class="ref" href="waybel_1.html#T9">WAYBEL_1:9</a></i>;<br/><a NAME="E3:57_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">g2</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">f2</font> is <a href="orders_3.html#V5">monotone</a> )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:57"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_7.html#T4" target="_self">Th4</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">s</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lattice3.html#K7">~</a> </span>)</span>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="lattice3.html#K7">~</a> </span>)</span>;<br/><a NAME="E1:57_1_1"/>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">t</font></span>)</span> &amp; (  <a href="lattice3.html#K9">~</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">s</font></span>)</span> implies <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">t</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">s</font> ) &amp; ( <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">t</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">s</font> implies  <a href="lattice3.html#K9">~</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">s</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:57_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_1.html#T9">WAYBEL_1:9</a></i>;<br/><b>hence </b><a NAME="E2:57_1_1"/>
( <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font> iff <font color="Maroon">t</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:57"><i><font color="Green">E32</font></i></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">[</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">f2</font></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 <b>by </b><i><a class="ref" href="yellow_7.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_1.html#T9">WAYBEL_1:9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:57"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">f2</font></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 ;<br/>

<a NAME="E4:57"/><b>then </b>
( <font color="Maroon">g2</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">f2</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lattice3.html#K7">~</a> </span>)</span><br/> for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="lattice3.html#K7">~</a> </span>)</span> holds <br/> ( <font color="Olive">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> iff <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">t</font> ) ) )
 
<b>by </b><i><a class="ref" href="waybel_1.html#T9">WAYBEL_1:9</a></i>;<br/>
<a NAME="E5:57"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">g1</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">f1</font> is <a href="orders_3.html#V5">monotone</a> )
 
<b>by </b><i><a class="txt" href="yellow_7.html#E1:57"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_7.html#T4" target="_self">Th4</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">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><a NAME="E1:57_2"/>
( <font color="Maroon">s</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">t</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="lattice3.html#K8">~</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="lattice3.html#K8">~</a> </span>)</span> &amp;  <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="lattice3.html#K8">~</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="lattice3.html#K8">~</a> </span>)</span> &amp; ( <font color="Maroon">s</font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="lattice3.html#K8">~</a> </span>)</span> implies <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t</font> <a href="lattice3.html#K8">~</a>  ) &amp; ( <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t</font> <a href="lattice3.html#K8">~</a>  implies <font color="Maroon">s</font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="lattice3.html#K8">~</a> </span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="waybel_1.html#T9">WAYBEL_1:9</a></i>;<br/><b>hence </b><a NAME="E2:57_2"/>
( <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> iff <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:57"><i><font color="Green">E32</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:57"/>
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">g1</font></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 <b>by </b><i><a class="ref" href="yellow_7.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_1.html#T9">WAYBEL_1:9</a></i>;<br/>


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