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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="unialg_1.html#V3">strict</a> <a href="unialg_2.html#V2">with_const_op</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<a NAME="E1:25"/><i><font color="Green">E16</font></i>: 
 <a href="unialg_2.html#K12">UnSubAlLattice</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V13">lower-bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12">UnSubAlLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12">UnSubAlLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<a NAME="E3:25_1"/>
 <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T19">UNIALG_2:19</a></i>;<br/>
<a NAME="E4:25_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E9:25_1"/>
 <a href="unialg_2.html#K7">GenUnivAlg</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E4:25_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="unialg_2.html#T22">UNIALG_2:22</a></i>;<br/>
<a NAME="E10:25_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="unialg_2.html#K8">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T23">UNIALG_2:23</a></i>;<br/>
<a NAME="E11:25_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_3.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E12:25_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<b>hence </b><a NAME="E13:25_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>

<b>thus </b><a NAME="E14:25_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E12:25_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25"/>
 <a href="unialg_2.html#K12">UnSubAlLattice</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V14">upper-bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<a NAME="E1:25_2"/>
<font color="Maroon">c<sub>1</sub></font> is  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T11">UNIALG_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12">UnSubAlLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12">UnSubAlLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#T11">UNIALG_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E8:25_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="unialg_2.html#K8">"\/"</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="unialg_3.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="unialg_2.html#K7">GenUnivAlg</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="unialg_2.html#D14">UNIALG_2:def 14</a></i>
<br/>.= 
 <a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_3.html#E8:25_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="unialg_2.html#T21">UNIALG_2:21</a></i>
;<br/>

<b>hence </b><a NAME="E10:25_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
 <a href="unialg_2.html#K12">UnSubAlLattice</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V15">bounded</a>
 <b>by </b><i><a class="txt" href="unialg_3.html#E1:25"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>


</div>
