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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</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">L'</font> being   <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">SubLattice</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">L'</font> is <a href="msualg_7.html#V2" title="MSUALG_7:attr.2">\/-inheriting</a> holds <br/>for <font color="Olive" title="b2">A'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">L'</font> holds   <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b2">A'</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b2">A'</font>,<font color="Olive" title="b1">L'</font></span><br/><b>let </b><font color="Maroon" title="c2">L'</font> be    <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">SubLattice</a> of <font color="Maroon" title="c1">L</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">L'</font> is <a href="msualg_7.html#V2" title="MSUALG_7:attr.2">\/-inheriting</a> implies for <font color="Olive" title="b1">A'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L'</font> holds   <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L'</font> )</span><br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="msualg_7.html#V2" title="MSUALG_7:attr.2">\/-inheriting</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">A'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L'</font> holds   <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L'</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">L'1</font> = <font color="Maroon" title="c2">L'</font> as  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">SubLattice</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="msualg_7.html#T16" target="_self" title="MSUALG_7:th.16">Th16</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">A'</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c2">L'</font></span><br/>

<b>set </b><font color="Maroon" title="c5">a</font> =  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c1">L</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a'</font> =  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c1">L</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L'1</font> <b>by </b><i><a class="txt" href="msualg_7.html#E1:24"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="msualg_7.html#D3" target="_self" title="MSUALG_7:def.3">Def3</a></i>;<br/>
<a NAME="E4:24"/>
( <font color="Maroon" title="c4">A'</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c1">L</font> &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c4">A'</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">c</font> holds <br/> <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">c</font> ) )
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E5:24"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c4">A'</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c6">a'</font>
 
<b>by </b><i><a class="ref" href="msualg_7.html#T14" target="_self" title="MSUALG_7:th.14">Th14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L'1</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">A'</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c7">c'</font> implies <font color="Maroon" title="c6">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c'</font> )</span><br/><a NAME="E1:24_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L'1</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="c1">L</font>
 <b>by </b><i><a class="ref" href="nat_lat.html#D16" title="NAT_LAT:def.16">NAT_LAT:def 16</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c7">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>assume </b><a NAME="E3:24_1"/>
<font color="Maroon" title="c4">A'</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c7">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c'</font></span><br/><a NAME="E4:24_1"/><b>then </b>
<font color="Maroon" title="c4">A'</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T14" target="_self" title="MSUALG_7:th.14">Th14</a></i>;<br/><a NAME="E5:24_1"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/><font color="Maroon" title="c6">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">c'</font> = 
<span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">c</font>
<b>by </b><i><a class="ref" href="msualg_7.html#T12" target="_self" title="MSUALG_7:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c6">a'</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E5:24_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>
;<br/><b>hence </b><a NAME="E7:24_1"/>
<font color="Maroon" title="c6">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</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>hence </b><a NAME="E7:24"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c2">L'</font>
 <b>by </b><i><a class="txt" href="msualg_7.html#E5:24"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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