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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V1">with_suprema</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:111"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K11" target="_self">finsups</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="waybel_0.html#K11" target="_self">finsups</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" target="_self">finsups</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16" target="_self">Th16</a>, <a class="ref" href="waybel_0.html#T50" target="_self">Th50</a></i>;<br/>
<b>hence </b><a NAME="E2:111"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" target="_self">finsups</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:111"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:111"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11" target="_self">finsups</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:111"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11" target="_self">finsups</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E4:111"><i><font color="Green">E53</font></i></a>, <a class="ref" href="waybel_0.html#D15" target="_self">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:111"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E7:111"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E12:111"/><i><font color="Green">E56</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a>, <a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:111_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:111_1"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="waybel_0.html#E12:111"><i><font color="Green">E56</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/><b>hence </b><a NAME="E3:111_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel_0.html#E12:111"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel_0.html#D19" target="_self">Def19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:111"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="txt" href="waybel_0.html#E3:111"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:111"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_0.html#E9:111"><i><font color="Green">E55</font></i></a>, <a class="txt" href="waybel_0.html#E13:111"><i><font color="Green">E57</font></i></a>, <a class="ref" href="waybel_0.html#T42" target="_self">Th42</a></i>;<br/>
<b>hence </b><a NAME="E16:111"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel_0.html#E7:111"><i><font color="Green">E54</font></i></a>, <a class="ref" href="waybel_0.html#D19" target="_self">Def19</a></i>;<br/>


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