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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</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">U0</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font> holds   <a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Olive" title="b1">U0</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a></span><br/><b>let </b><font color="Maroon" title="c2">U0</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a></span><br/>



<b>set </b><font color="Maroon" title="c3">L</font> =  <a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font>;<br/>
<b>thus </b><a NAME="E1:70"/>
 <a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c4">C</font> =  <a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self" title="MSUALG_2:def.20">Def20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">G1</font> = <font color="Maroon" title="c5">G</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">G1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span> holds <br/> ( <font color="Maroon" title="c6">G1</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> )</span><br/>

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">G1</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> &amp; <font color="Maroon" title="c7">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c7">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a2</font> = <font color="Maroon" title="c8">a1</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self" title="MSUALG_2:def.20">Def20</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c6">G1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">a</font> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <span class="p2">(<span class="default"><a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <font color="Maroon" title="c2">U0</font></span>)</span></span>)</span> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c9">a2</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D22" target="_self" title="MSUALG_2:def.22">Def22</a></i>
<br/>.= 
<font color="Maroon" title="c6">G1</font>
<b>by </b><i><a class="ref" href="msualg_2.html#T24" target="_self" title="MSUALG_2:th.24">Th24</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font></span><br/>

<b>hence </b><a NAME="E6:70_1"/>
<font color="Maroon" title="c7">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font>
 ; <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="E2:70"/>
 <a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
  <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="70_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c4">B</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U0</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self" title="MSUALG_2:def.20">Def20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">G1</font> = <font color="Maroon" title="c5">G</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">G1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span> holds <br/> ( <font color="Maroon" title="c6">G1</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> )</span><br/>

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <font color="Maroon" title="c2">U0</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">G1</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> &amp; <font color="Maroon" title="c7">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c7">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon" title="c2">U0</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a2</font> = <font color="Maroon" title="c8">a1</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self" title="MSUALG_2:def.20">Def20</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c6">G1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">a</font> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c9">a2</font>
<b>by </b><i><a class="ref" href="msualg_2.html#D21" target="_self" title="MSUALG_2:def.21">Def21</a></i>
<br/>.= 
<font color="Maroon" title="c6">G1</font>
<b>by </b><i><a class="ref" href="msualg_2.html#T26" target="_self" title="MSUALG_2:th.26">Th26</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font></span><br/>

<b>hence </b><a NAME="E7:70_2"/>
<font color="Maroon" title="c7">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font>
 ; <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>


</div>
