<?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#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">g</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">d</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>assume </b><a NAME="E1:43"/><i><font color="Green" title="E32">A1</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">g</font>,<font color="Maroon" title="c4">d</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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon" title="c3">g</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 ;<br/><a NAME="E2:43_1"/><b>then </b>
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> holds  <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="waybel_1.html#T23" target="_self" title="WAYBEL_1:th.23">Th23</a></i>;<br/><a NAME="E3:43_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T24" target="_self" title="WAYBEL_1:th.24">Th24</a></i>;<br/><b>hence </b><a NAME="E4:43_1"/>
<font color="Maroon" title="c4">d</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T25" target="_self" title="WAYBEL_1:th.25">Th25</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:43"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c4">d</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<a NAME="E4:43"/><i><font color="Green" title="E34">A3</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">d</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">d</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green" title="E35">A4</font></i>: 
( <font color="Maroon" title="c3">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c4">d</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="waybel_1.html#T9" target="_self" title="WAYBEL_1:th.9">Th9</a></i>;<br/>
<a NAME="E6:43"/>
( <font color="Maroon" title="c4">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c2">T</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">d</font> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:43"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="waybel_1.html#T19" target="_self" title="WAYBEL_1:th.19">Th19</a></i>;<br/>
<b>then </b><font color="Maroon" title="c4">d</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="txt" href="waybel_1.html#E5:43"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="waybel_1.html#T21" target="_self" title="WAYBEL_1:th.21">Th21</a></i>
<br/>.= 
<font color="Maroon" title="c4">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">d</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
;<br/>
<b>then </b><font color="Maroon" title="c3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">d</font> = 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:43"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="waybel_1.html#E4:43"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_1.html#T50" title="FUNCT_1:th.50">FUNCT_1:50</a></i>
<br/>.= 
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="ref" href="grcat_1.html#D11" title="GRCAT_1:def.11">GRCAT_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E9:43"/>
<font color="Maroon" title="c3">g</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T25" target="_self" title="WAYBEL_1:th.25">Th25</a></i>;<br/>


</div>
