<?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>, <font color="Olive" title="b3">C</font> being   <a href="interva1.html#M2" title="INTERVA1:mode.2">RoughSet</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Olive" title="b3">C</font> <a href="interva1.html#R2" title="INTERVA1:pred.2">=</a> <font color="Olive" title="b1">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Olive" title="b3">C</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font> <a href="interva1.html#R2" title="INTERVA1:pred.2">=</a> <font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span></span><br/>







<a NAME="E1:82"/><span class="lab"><font color="Green" title="E59">a</font></span>: 
<span class="p1">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c3">B</font></span>)</span></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="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c3">B</font></span>)</span></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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T56" target="_self" title="INTERVA1:th.56">Th1</a></span>;<br/>
<a NAME="E2:82"/><span class="lab"><font color="Green" title="E60">s</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c3">B</font></span>)</span></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="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<a NAME="E3:82"/><span class="lab"><font color="Green" title="E61">d</font></span>: 
<span class="p1">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c2">A</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T56" target="_self" title="INTERVA1:th.56">Th1</a></span>;<br/>
<a NAME="E4:82"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
<span class="p1">(<span class="default"><a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <font color="Maroon" title="c2">A</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T56" target="_self" title="INTERVA1:th.56">Th1</a></span>;<br/>
<a NAME="E5:82"/><span class="lab"><font color="Green" title="E63">T1</font></span>: 
 <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K14" title="INTERVA1:func.14">LAp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>
 
<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="txt" href="#E4:82"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E59">a</font></span></a>, <a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E60">s</font></span></a>, <a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E61">d</font></span></a></span>;<br/>
<a NAME="E6:82"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
 <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c3">B</font></span>)</span></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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T57" target="_self" title="INTERVA1:th.57">Th2</a></span>;<br/>
<a NAME="E7:82"/><span class="lab"><font color="Green" title="E65">q</font></span>: 
<span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c3">B</font></span>)</span></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="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c3">B</font></span>)</span></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="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T57" target="_self" title="INTERVA1:th.57">Th2</a></span>;<br/>
<a NAME="E8:82"/><span class="lab"><font color="Green" title="E66">w</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c3">B</font></span>)</span></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="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<a NAME="E9:82"/><span class="lab"><font color="Green" title="E67">e</font></span>: 
<span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c2">A</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T57" target="_self" title="INTERVA1:th.57">Th2</a></span>;<br/>
<a NAME="E10:82"/>
<span class="p1">(<span class="default"><a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <font color="Maroon" title="c2">A</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K15" title="INTERVA1:func.15">UAp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T57" target="_self" title="INTERVA1:th.57">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:82"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font> <a href="interva1.html#R2" title="INTERVA1:pred.2">=</a> <font color="Maroon" title="c2">A</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="interva1.html#K16" title="INTERVA1:func.16">_\/_</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E63">T1</font></span></a>, <a class="ref" href="interva1.html#D17" target="_self" title="INTERVA1:def.17">Def5</a>, <a class="txt" href="#E6:82"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E65">q</font></span></a>, <a class="txt" href="#E8:82"><span class="lab"><font color="Green" title="E66">w</font></span></a>, <a class="txt" href="#E9:82"><span class="lab"><font color="Green" title="E67">e</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>
