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<b>let </b><font color="Maroon" title="c1">U0</font> be  <a href="unialg_2.html#V2" title="UNIALG_2:attr.2">with_const_op</a> <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">l1</font>, <font color="Maroon" title="c3">l2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">U1</font>, <font color="Maroon" title="c5">U2</font> be  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>;<br/>





<b>reconsider </b><font color="Maroon" title="c6">l1</font> = <font color="Maroon" title="c4">U1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> <b>by </b><i><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">l2</font> = <font color="Maroon" title="c5">U2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> <b>by </b><i><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></i>;<br/>
<a NAME="E3:23"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c6">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">l2</font> implies the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon" title="c6">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">l2</font>
 ;<br/>

<a NAME="E2:23_1"/><b>then </b>
<font color="Maroon" title="c6">l1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l2</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/>
<a NAME="E3:23_1"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c4">U1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c5">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D16" title="UNIALG_2:def.16">UNIALG_2:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">U11</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">U21</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">U3</font> = <font color="Maroon" title="c8">U11</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">U21</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E7:23_1"/>
 <a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c10">U3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">U2</font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E3:23_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="unialg_2.html#D14" title="UNIALG_2:def.14">UNIALG_2:def 14</a></i>;<br/>
<a NAME="E8:23_1"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D13" title="UNIALG_2:def.13">UNIALG_2:def 13</a></i>;<br/>
<a NAME="E9:23_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:23_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E8:23_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E11:23_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:23"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font> implies <font color="Maroon" title="c6">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">l2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font>
 ;<br/>

<a NAME="E2:23_2"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">U11</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">U21</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">U3</font> = <font color="Maroon" title="c8">U11</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">U21</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E6:23_2"/>
 <a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c10">U3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">U2</font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E2:23_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="unialg_2.html#T22" title="UNIALG_2:th.22">UNIALG_2:22</a></i>;<br/>
<a NAME="E7:23_2"/><b>then </b>
<font color="Maroon" title="c4">U1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c5">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D14" title="UNIALG_2:def.14">UNIALG_2:def 14</a></i>;<br/>
<a NAME="E8:23_2"/><b>then </b>
<font color="Maroon" title="c6">l1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D16" title="UNIALG_2:def.16">UNIALG_2:def 16</a></i>;<br/>
<b>hence </b><a NAME="E9:23_2"/>
<font color="Maroon" title="c6">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">l2</font>
 <b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:23"/>
( <font color="Maroon" title="c2">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">U1</font> &amp; <font color="Maroon" title="c3">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">U2</font> implies ( <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font> iff the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">U2</font> ) )
 <b>by </b><i><a class="txt" href="unialg_3.html#E3:23"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>


</div>
