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

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>( non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font>, non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font>) <span class="kw">-&gt; </span>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font> = $1 <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> $2;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>( non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font>, non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font>) <span class="kw">-&gt; </span>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font> = $1 <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> $2;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">B</font> =  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>] <span class="kw">means </span> ex <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font> st <br/>( <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Olive" title="b2">b9</font> );<br/>
<a NAME="E1:72_1_1"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>  ex <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">x9</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y9</font> = <font color="Maroon" title="c4">y</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D11" target="_self" title="INTERVA1:def.11">DefAG</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">z</font> = <font color="Maroon" title="c5">x9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c6">y9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D11" target="_self" title="INTERVA1:def.11">DefAG</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">z</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">z</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:72_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">z</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">consider </span><font color="Maroon" title="c3">B1</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:72_1_1"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c3">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E1:72_1_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>] <span class="kw">means </span> ex <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font> st <br/>( <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Olive" title="b2">b9</font> );<br/>
<a NAME="E4:72_1_1"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>  ex <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">x9</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y9</font> = <font color="Maroon" title="c5">y</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D11" target="_self" title="INTERVA1:def.11">DefAG</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c6">x9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c7">y9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D11" target="_self" title="INTERVA1:def.11">DefAG</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">z</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">S<sub>2</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">z</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:72_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">z</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">consider </span><font color="Maroon" title="c4">B2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:72_1_1"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c4">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E4:72_1_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>);<br/></span>
<span class="kw">take </span><font color="Maroon" title="c5">IT</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font></span>)</span>,<font color="Maroon" title="c3">B1</font>,<font color="Maroon" title="c4">B2</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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 <font color="Maroon" title="c5">IT</font><br/> for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font>  st <font color="Olive" title="b3">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/>( the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Olive" title="b4">b9</font> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Olive" title="b4">b9</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:72_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font>
 ; <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="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 <font color="Maroon" title="c5">IT</font><br/> for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font>  st <font color="Olive" title="b3">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/>( the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Olive" title="b4">b9</font> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Olive" title="b4">b9</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be    <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 <font color="Maroon" title="c5">IT</font>; <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 non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font>  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> holds <br/>( the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Olive" title="b2">b9</font> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Olive" title="b2">b9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">a9</font>, <font color="Maroon" title="c9">b9</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</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="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> implies ( the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c9">b9</font> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c9">b9</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E8:72_1_1"/><span class="lab"><font color="Green" title="E53">F0</font></span>: 
( <font color="Maroon" title="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c9">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c9">b9</font> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c9">b9</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c7">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">a9</font>, <font color="Maroon" title="c13">b9</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font><span class="kw"> such that </span><br/><a NAME="E11:72_1_1"/><span class="lab"><font color="Green" title="E54">F1</font></span>: 
( <font color="Maroon" title="c12">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c13">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c3">B1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c13">b9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_1_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">a1</font>, <font color="Maroon" title="c15">b1</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#M1" title="INTERVA1:mode.1">IntervalSet</a> of <font color="Maroon" title="c1">U</font><span class="kw"> such that </span><br/><a NAME="E13:72_1_1"/><span class="lab"><font color="Green" title="E55">F2</font></span>: 
( <font color="Maroon" title="c14">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c15">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c15">b1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E14:72_1_1"/>
( the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c9">b9</font> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c5">IT</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c9">b9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72_1_1"><span class="lab"><font color="Green" title="E54">F1</font></span></a>, <a class="txt" href="#E13:72_1_1"><span class="lab"><font color="Green" title="E55">F2</font></span></a>, <a class="txt" href="#E8:72_1_1"><span class="lab"><font color="Green" title="E53">F0</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/>


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