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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">g</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">d</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">d</font> is <a href="orders_3.html#V5">monotone</a>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">d</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="yellow_2.html#R1">&lt;=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green">E48</font></i>: 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E5:37"/>
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">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> iff <font color="Maroon">d</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </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/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1_1"/>
<font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:37_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:37_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T2" target="_self">Th2</a>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/><a NAME="E4:37_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E5:37_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>;<br/><b>hence </b><a NAME="E6:37_1_1"/>
<font color="Maroon">d</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="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:37_1"/>
<font color="Maroon">d</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>
 ;<br/>

<a NAME="E3:37_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:37_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>
<a NAME="E5:37_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E6:37_1"/><b>then </b>
<font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>;<br/>
<b>hence </b><a NAME="E7:37_1"/>
<font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:37"/>
<span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 <b>by </b><i>, , </i>;<br/>


</div>
