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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font> being  <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/> for <font color="Olive" title="b3">U1</font>, <font color="Olive" title="b4">U2</font> being <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>  st <font color="Olive" title="b1">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">U1</font> &amp; <font color="Olive" title="b2">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">U2</font> holds <br/>( <font color="Olive" title="b1">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">l2</font> iff <font color="Olive" title="b3">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Olive" title="b4">U2</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">U1</font>, <font color="Olive" title="b2">U2</font> being <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>  st <font color="Maroon" title="c2">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U1</font> &amp; <font color="Maroon" title="c3">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">U2</font> holds <br/>( <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 <font color="Olive" title="b1">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Olive" title="b2">U2</font> )</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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 <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font> ) )</span><br/>





<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E14">A1</font></i>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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 <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font> )</span><br/>

<b>thus </b><a NAME="E2:24"/>
( <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font> implies <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font></span><br/>

<a NAME="E2:24_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="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="#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="unialg_3.html#T17" target="_self" title="UNIALG_3:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1"/>
<font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#T14" title="UNIALG_2:th.14">UNIALG_2:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:24"/>
( <font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_2"/>
<font color="Maroon" title="c4">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c5">U2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font></span><br/>

<a NAME="E2:24_2"/><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> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">U2</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>hence </b><a NAME="E3:24_2"/>
<font color="Maroon" title="c2">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">l2</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="unialg_3.html#T17" target="_self" title="UNIALG_3:th.17">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


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