<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="roughs_1.html#NM2" title="ROUGHS_1:NM.2">Tolerance_Space</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" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <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><br/> for <font color="Olive" title="b3">A9</font>, <font color="Olive" title="b4">B9</font> being   <a href="interva1.html#M2" title="INTERVA1:mode.2">RoughSet</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A9</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B9</font> holds <br/>( <font color="Olive" title="b1">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">B</font> iff (  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Olive" title="b3">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Olive" title="b4">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Olive" title="b3">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Olive" title="b4">B9</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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"> for <font color="Olive" title="b1">A9</font>, <font color="Olive" title="b2">B9</font> being   <a href="interva1.html#M2" title="INTERVA1:mode.2">RoughSet</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A9</font> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B9</font> holds <br/>( <font color="Maroon" title="c2">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">B</font> iff (  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Olive" title="b1">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Olive" title="b2">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Olive" title="b1">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Olive" title="b2">B9</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A9</font>, <font color="Maroon" title="c5">B9</font> be    <a href="interva1.html#M2" title="INTERVA1:mode.2">RoughSet</a> of <font color="Maroon" title="c1">X</font>; <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="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A9</font> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B9</font> implies ( <font color="Maroon" title="c2">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">B</font> iff (  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:108"/><span class="lab"><font color="Green" title="E77">Z1</font></span>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A9</font> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B9</font> )
 ; <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="c2">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">B</font> iff (  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> ) )</span><br/>

<a NAME="E2:108"/><span class="lab"><font color="Green" title="E78">Z2</font></span>: 
( <font color="Maroon" title="c2">A</font> is    <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> &amp; <font color="Maroon" title="c3">B</font> is    <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">thus </span><a NAME="E3:108"/>
( <font color="Maroon" title="c2">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">B</font> implies (  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> ) )
  <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#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> implies <font color="Maroon" title="c2">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">B</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:108_1"/>
<font color="Maroon" title="c2">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">B</font>
 ; <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#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> )</span><br/>

<a NAME="E2:108_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>;<br/>
<a NAME="E3:108_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A9</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c5">B9</font> <a href="interva1.html#R2" title="INTERVA1:pred.2">=</a> <font color="Maroon" title="c5">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E77">Z1</font></span></a>, <a class="ref" href="interva1.html#D23" target="_self" title="INTERVA1:def.23">Def8</a>, <a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E78">Z2</font></span></a></span>;<br/>
<a NAME="E4:108_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">A9</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp; <span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">A9</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T56" target="_self" title="INTERVA1:th.56">Th1</a>, <a class="ref" href="interva1.html#T57" target="_self" title="INTERVA1:th.57">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:108_1"/>
(  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></span>; <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">assume </span><a NAME="E4:108"/>
(  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> )
 ; <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="c2">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">B</font></span><br/>

<a NAME="E5:108"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">A9</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp; <span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">A9</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E6:108"/><span class="kw">then </span>
(  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A9</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c5">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c5">B9</font> &amp;  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A9</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c5">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c5">B9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T56" target="_self" title="INTERVA1:th.56">Th1</a>, <a class="ref" href="interva1.html#T57" target="_self" title="INTERVA1:th.57">Th2</a></span>;<br/>
<a NAME="E7:108"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<font color="Maroon" title="c4">A9</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c5">B9</font> <a href="interva1.html#R2" title="INTERVA1:pred.2">=</a> <font color="Maroon" title="c5">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D17" target="_self" title="INTERVA1:def.17">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">A1</font> = <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c7">B1</font> = <font color="Maroon" title="c3">B</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="c8">A9</font> = <font color="Maroon" title="c6">A1</font>, <font color="Maroon" title="c9">B9</font> = <font color="Maroon" title="c7">B1</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/>
<a NAME="E10:108"/>
<font color="Maroon" title="c8">A9</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c9">B9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">B</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">hence </span><a NAME="E11:108"/>
<font color="Maroon" title="c2">A</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a>, <a class="txt" href="#E7:108"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E77">Z1</font></span></a></span>; <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>
