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

<a NAME="E1:73_1_1"/><span class="lab"><font color="Green" title="E50">a0</font></span>: 
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:73_1_1_1"/>
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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</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="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E1:73_1_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> )
 
;<br/>
<a NAME="E2:73_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a9</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c5">b9</font> = <font color="Maroon" title="c3">b</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/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font> = 
<font color="Maroon" title="c4">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c5">b9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
<br/>.= 
<font color="Maroon" title="c5">b9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c4">a9</font>

<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:73_1_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">a</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">hence </span><a NAME="E2:73_1_1_1"/>
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</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>
<a NAME="E2:73_1_1"/><span class="lab"><font color="Green" title="E51">a1</font></span>: 
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:73_1_1_2"/>
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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</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="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font></span><br/>

<a NAME="E1:73_1_1_2_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> )
 
;<br/>
<a NAME="E2:73_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">a9</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">b9</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c7">c9</font> = <font color="Maroon" title="c4">c</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">bc</font> = <font color="Maroon" title="c6">b9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c7">c9</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> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">ab</font> = <font color="Maroon" title="c5">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c6">b9</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> ;<br/>
<a NAME="E6:73_1_1_2_1"/>
( <font color="Maroon" title="c8">bc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c9">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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/>
<a NAME="E7:73_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">Y</font></span>: 
( <font color="Maroon" title="c8">bc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c9">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">bc</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
<br/>.= 
<font color="Maroon" title="c5">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c8">bc</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a>, <a class="txt" href="#E7:73_1_1_2_1"><span class="lab"><font color="Green" title="E52">Y</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c6">b9</font></span>)</span> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c7">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T23" target="_self" title="INTERVA1:th.23">Asso1</a></span>
<br/>.= 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">ab</font>,<font color="Maroon" title="c7">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a>, <a class="txt" href="#E7:73_1_1_2_1"><span class="lab"><font color="Green" title="E52">Y</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:73_1_1_2_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</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">hence </span><a NAME="E2:73_1_1_2"/>
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</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>
<a NAME="E3:73_1_1"/><span class="lab"><font color="Green" title="E52">a2</font></span>: 
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:73_1_1_3"/>
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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span>; <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="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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><br/>

<a NAME="E1:73_1_1_3_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> )
 
;<br/>
<a NAME="E2:73_1_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a9</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c5">b9</font> = <font color="Maroon" title="c3">b</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="c6">ab</font> = <font color="Maroon" title="c4">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c5">b9</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> ;<br/>
<a NAME="E5:73_1_1_3_1"/>
<font color="Maroon" title="c6">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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/>
<a NAME="E6:73_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">Y</font></span>: 
<font color="Maroon" title="c6">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">ab</font>,<font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
<br/>.= 
<font color="Maroon" title="c6">ab</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c5">b9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a>, <a class="txt" href="#E6:73_1_1_3_1"><span class="lab"><font color="Green" title="E53">Y</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T36" target="_self" title="INTERVA1:th.36">Abs2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:73_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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>
 ; <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">hence </span><a NAME="E2:73_1_1_3"/>
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</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>
<a NAME="E4:73_1_1"/><span class="lab"><font color="Green" title="E53">a3</font></span>: 
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:73_1_1_4"/>
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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</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="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</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"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span><br/>

<a NAME="E1:73_1_1_4_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> )
 
;<br/>
<a NAME="E2:73_1_1_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">a9</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">b9</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c7">c9</font> = <font color="Maroon" title="c4">c</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">bc</font> = <font color="Maroon" title="c6">b9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c7">c9</font>, <font color="Maroon" title="c9">ab</font> = <font color="Maroon" title="c5">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c6">b9</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> ;<br/>
<a NAME="E5:73_1_1_4_1"/>
( <font color="Maroon" title="c8">bc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c9">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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/>
<a NAME="E6:73_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">Y</font></span>: 
( <font color="Maroon" title="c8">bc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c9">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">bc</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
<br/>.= 
<font color="Maroon" title="c5">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c8">bc</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a>, <a class="txt" href="#E6:73_1_1_4_1"><span class="lab"><font color="Green" title="E54">Y</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c6">b9</font></span>)</span> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c7">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T24" target="_self" title="INTERVA1:th.24">Prz1</a></span>
<br/>.= 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">ab</font>,<font color="Maroon" title="c7">c9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a>, <a class="txt" href="#E6:73_1_1_4_1"><span class="lab"><font color="Green" title="E54">Y</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:73_1_1_4_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</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"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</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">hence </span><a NAME="E2:73_1_1_4"/>
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</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>
<a NAME="E5:73_1_1"/><span class="lab"><font color="Green" title="E54">a4</font></span>: 
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:73_1_1_5"/>
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 <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</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="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E1:73_1_1_5_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> )
 
;<br/>
<a NAME="E2:73_1_1_5_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a9</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c5">b9</font> = <font color="Maroon" title="c3">b</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="c6">ab</font> = <font color="Maroon" title="c4">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c5">b9</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> ;<br/>
<a NAME="E5:73_1_1_5_1"/>
<font color="Maroon" title="c6">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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/>
<a NAME="E6:73_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">X</font></span>: 
<font color="Maroon" title="c6">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">a9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a9</font> <a href="interva1.html#K4" title="INTERVA1:func.4">_\/_</a> <font color="Maroon" title="c5">b9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
<br/>.= 
<font color="Maroon" title="c4">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c6">ab</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a>, <a class="txt" href="#E6:73_1_1_5_1"><span class="lab"><font color="Green" title="E55">X</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T35" target="_self" title="INTERVA1:th.35">Abs1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:73_1_1_5_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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">hence </span><a NAME="E2:73_1_1_5"/>
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</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>
<a NAME="E6:73_1_1"/>
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_6"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:73_1_1_6"/>
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 <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</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="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</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> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E1:73_1_1_6_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> )
 
;<br/>
<a NAME="E2:73_1_1_6_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="interva1.html#K11" title="INTERVA1:func.11">IntervalSets</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a9</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c5">b9</font> = <font color="Maroon" title="c3">b</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/>
<a NAME="E4:73_1_1_6_1"/>
<font color="Maroon" title="c4">a9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c5">b9</font> <a href="interva1.html#R1" title="INTERVA1:pred.1">=</a> <font color="Maroon" title="c5">b9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c4">a9</font>
 
;<br/>
<span class="kw">then </span>the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> = 
<font color="Maroon" title="c5">b9</font> <a href="interva1.html#K3" title="INTERVA1:func.3">_/\_</a> <font color="Maroon" title="c4">a9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
<br/>.= 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#D12" target="_self" title="INTERVA1:def.12">DefLLL</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:73_1_1_6_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</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> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">a</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">hence </span><a NAME="E2:73_1_1_6"/>
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</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">hence </span><a NAME="E7:73_1_1"/>
 <a href="interva1.html#K12" title="INTERVA1:func.12">InterLatt</a> <font color="Maroon" title="c1">U</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1"><span class="lab"><font color="Green" title="E50">a0</font></span></a>, <a class="txt" href="#E2:73_1_1"><span class="lab"><font color="Green" title="E51">a1</font></span></a>, <a class="txt" href="#E3:73_1_1"><span class="lab"><font color="Green" title="E52">a2</font></span></a>, <a class="txt" href="#E4:73_1_1"><span class="lab"><font color="Green" title="E53">a3</font></span></a>, <a class="txt" href="#E5:73_1_1"><span class="lab"><font color="Green" title="E54">a4</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>
