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<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/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1"/>
<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>
 ;<br/><b>then consider </b><font color="Maroon">g'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>, <font color="Maroon">d'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green">E32</font></i>: 
<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> <a href="hidden.html#R1">=</a> <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>
 <b>and </b><br/><a NAME="E4:20_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">g'</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">d'</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">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> ) ) )
 <b>by </b><i/>;<br/><a NAME="E5:20_1"/>
( <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d'</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E6:20_1"/>
( <font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">d</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">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> ) ) )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:20"/>
( <font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">d</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">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> ) ) implies <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/>;<br/>


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