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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:43"/><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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_2.html#V2">onto</a>
 ;<br/><a NAME="E2:43_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="waybel_1.html#R3" target="_self">is_minimum_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_1.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E3:43_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E4:43_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T25" target="_self">Th25</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:43"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<a NAME="E4:43"/><i><font color="Green">E34</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="orders_3.html#V5">monotone</a> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E6:43"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow_2.html#R1">&lt;=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_1.html#T19" target="_self">Th19</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="waybel_1.html#E5:43"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_1.html#T21" target="_self">Th21</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/>
<b>then </b><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:43"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel_1.html#E4:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T50">FUNCT_1:50</a></i>
<br/>.= 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E9:43"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T25" target="_self">Th25</a></i>;<br/>


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