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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:98_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E3:98_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D4">YELLOW_2:def 4</a></i>;<br/><a NAME="E4:98_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T1">YELLOW_2:1</a></i>;<br/><a NAME="E5:98_1_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><b>hence </b><a NAME="E6:98_1_1"/>
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E3:98_1"/>
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E4:98_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>;<br/><a NAME="E5:98_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><a NAME="E6:98_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E4:98_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="yellow_0.html#T4">YELLOW_0:4</a></i>;<br/><a NAME="E7:98_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T1">YELLOW_2:1</a></i>;<br/><a NAME="E8:98_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D4">YELLOW_2:def 4</a></i>;<br/><b>hence </b><a NAME="E9:98_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:98"/>
<span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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><a class="ref" href="waybel_1.html#T9" target="_self">Th9</a></i>;<br/>

<a NAME="E3:98"/><b>then </b>
 <a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V5" target="_self">lower_adjoint</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E4:98"/>
 <a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T14" target="_self">Th14</a></i>;<br/>


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