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<span class="kw">thus </span><a NAME="E1:107_1_1"/>
 <a href="interva1.html#K21" title="INTERVA1:func.21">RSLattice</a> <font color="Maroon" title="c1">X</font> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> 
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="interva1.html#K21" title="INTERVA1:func.21">RSLattice</a> <font color="Maroon" title="c1">X</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="107_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">E</font> =  <a href="interva1.html#K22" title="INTERVA1:func.22">ERS</a> <font color="Maroon" title="c1">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K18" title="INTERVA1:func.18">RoughSets</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="interva1.html#E106"><span class="lab"><font color="Green" title="E72">Th22</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">e</font> = <font color="Maroon" title="c2">E</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="interva1.html#K21" title="INTERVA1:func.21">RSLattice</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D23" target="_self" title="INTERVA1:def.23">Def8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">e</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="interva1.html#K21" title="INTERVA1:func.21">RSLattice</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Maroon" title="c3">e</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="c3">e</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="interva1.html#K21" title="INTERVA1:func.21">RSLattice</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c4">u</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">K</font> = <font color="Maroon" title="c4">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K18" title="INTERVA1:func.18">RoughSets</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D23" target="_self" title="INTERVA1:def.23">Def8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">E9</font> = <font color="Maroon" title="c2">E</font>, <font color="Maroon" title="c7">K9</font> = <font color="Maroon" title="c5">K</font> as    <a href="interva1.html#M2" title="INTERVA1:mode.2">RoughSet</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D20" target="_self" title="INTERVA1:def.20">DefRSX</a></span>;<br/>
<font color="Maroon" title="c3">e</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">u</font> = 
<font color="Maroon" title="c6">E9</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c7">K9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D23" target="_self" title="INTERVA1:def.23">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T69" target="_self" title="INTERVA1:th.69">Th23</a></span>
;<br/>
<a NAME="E6:107_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">e</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c4">u</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:107_1_1_1"/>
( <font color="Maroon" title="c3">e</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c4">u</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:107_1_1"/>
 <a href="interva1.html#K21" title="INTERVA1:func.21">RSLattice</a> <font color="Maroon" title="c1">X</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">E</font> =  <a href="interva1.html#K23" title="INTERVA1:func.23">TRS</a> <font color="Maroon" title="c1">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K18" title="INTERVA1:func.18">RoughSets</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="interva1.html#E109"><span class="lab"><font color="Green" title="E74">Th24</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">e</font> = <font color="Maroon" title="c2">E</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="interva1.html#K21" title="INTERVA1:func.21">RSLattice</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D23" target="_self" title="INTERVA1:def.23">Def8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">e</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="interva1.html#K21" title="INTERVA1:func.21">RSLattice</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Maroon" title="c3">e</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="c3">e</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="interva1.html#K21" title="INTERVA1:func.21">RSLattice</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c4">u</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">K</font> = <font color="Maroon" title="c4">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K18" title="INTERVA1:func.18">RoughSets</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D23" target="_self" title="INTERVA1:def.23">Def8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">E9</font> = <font color="Maroon" title="c2">E</font>, <font color="Maroon" title="c7">K9</font> = <font color="Maroon" title="c5">K</font> as    <a href="interva1.html#M2" title="INTERVA1:mode.2">RoughSet</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D20" target="_self" title="INTERVA1:def.20">DefRSX</a></span>;<br/>
<font color="Maroon" title="c3">e</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">u</font> = 
<font color="Maroon" title="c6">E9</font> <a href="interva1.html#K17" title="INTERVA1:func.17">_/\_</a> <font color="Maroon" title="c7">K9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D23" target="_self" title="INTERVA1:def.23">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T70" target="_self" title="INTERVA1:th.70">Th25</a></span>
;<br/>
<a NAME="E6:107_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c3">e</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c4">u</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:107_1_1_2"/>
( <font color="Maroon" title="c3">e</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c4">u</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
