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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="lfuzzy_0.html#V2" target="_self">interval</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E2:24"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:24"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E6:24"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="integra1.html#V1">closed-interval</a>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:24"><i><font color="Green">E12</font></i></a>, <a class="ref" href="integra1.html#D1">INTEGRA1:def 1</a></i>;<br/>
<a NAME="E7:24"/><b>then </b><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="integra1.html#T3">INTEGRA1:3</a></i>;<br/>
<a NAME="E8:24"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E4:24"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rcomp_1.html#T4">RCOMP_1:4</a></i>;<br/>
<a NAME="E9:24"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:24"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/><b>thus </b><a NAME="E3:24_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:24_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow_0.html#T6">YELLOW_0:6</a></i>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E4:24_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:24"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>9</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2"/><i><font color="Green">E16</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:24_1_1_2"/>
 <a href="seq_4.html#K4">upper_bound</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:24"><i><font color="Green">E12</font></i></a>, <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/><a NAME="E3:24_1_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E4:24"><i><font color="Green">E13</font></i></a>, <a class="txt" href="lfuzzy_0.html#E7:24"><i><font color="Green">E14</font></i></a>, <a class="txt" href="lfuzzy_0.html#E1:24_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="seq_4.html#T65">SEQ_4:65</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E5:24_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:24_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E7:24_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E8:24"><i><font color="Green">E15</font></i></a>, <a class="txt" href="lfuzzy_0.html#E1:24_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><a NAME="E8:24_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E4:24"><i><font color="Green">E13</font></i></a>, <a class="txt" href="lfuzzy_0.html#E5:24_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:24_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E7:24_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:24"><i><font color="Green">E12</font></i></a>, <a class="txt" href="lfuzzy_0.html#E3:24_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/><a NAME="E11:24_1_1_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:24_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:24_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E8:24"><i><font color="Green">E15</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:24_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:24_1_1_2_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:24_1_1_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:24_1_1_2_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:24"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:24_1_1_2_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:24_1_1_2_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="lfuzzy_0.html#E1:24_1_1_2_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:24_1_1_2_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_1_1_2_2"/><b>then </b>
 <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:24_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="topmetr3.html#T1">TOPMETR3:1</a></i>;<br/>
<b>hence </b><a NAME="E4:24_1_1_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E13:24_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E11:24_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:24"/><b>then </b>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/>
<b>hence </b><a NAME="E11:24"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:24"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow_0.html#T50">YELLOW_0:50</a></i>;<br/>


</div>
