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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<i><font color="Green">E10</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> #)

<br/>.= 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E3:31"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_1.html#E1:31"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yellow_1.html#E1:31"><i><font color="Green">E10</font></i></a>, <a class="ref" href="setfam_1.html#T8">SETFAM_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:31_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/><b>hence </b><a NAME="E3:31_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:31"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E1:31_3"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_3_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E3:31_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="yellow_1.html#E1:31_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow_1.html#E1:31_3_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E4:31_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><a NAME="E3:31_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/><b>hence </b><a NAME="E4:31_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:31"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_1.html#E6:31"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>


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