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

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="lfuzzy_0.html#V1" title="LFUZZY_0:attr.1">real</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_0.html#D2" target="_self" title="LFUZZY_0:def.2">LFUZZY_0:def 2</a><br/>

<b>given </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><a NAME="E2:21_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/>

<a NAME="E3:21_1_1"/>
 ex <font color="Olive" title="b1">x</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">x</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 <font color="Maroon" title="c1">R</font> &amp; ( for <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">y</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 <font color="Maroon" title="c1">R</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c3">y</font>
;<br/>

<b>thus </b><a NAME="E1:21_1_1_1"/>
<font color="Maroon" title="c3">y</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 <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:21_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>

<b>let </b><font color="Maroon" title="c4">z</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E2:21_1_1_1"/>
<font color="Maroon" title="c4">z</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 <font color="Maroon" title="c1">R</font>
 ;<br/>

<b>hence </b><a NAME="E3:21_1_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:21_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21_1_1"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:21_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="lfuzzy_0.html#T7" target="_self" title="LFUZZY_0:th.7">Th7</a></i>;<br/>
<a NAME="E5:21_1_1"/>
 ex <font color="Olive" title="b1">x</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">x</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 <font color="Maroon" title="c1">R</font> &amp; ( for <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">y</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 <font color="Maroon" title="c1">R</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c2">x</font>
;<br/>

<b>thus </b><a NAME="E1:21_1_1_2"/>
<font color="Maroon" title="c2">x</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 <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:21_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>

<b>let </b><font color="Maroon" title="c4">z</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E2:21_1_1_2"/>
<font color="Maroon" title="c4">z</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 <font color="Maroon" title="c1">R</font>
 ;<br/>

<b>hence </b><a NAME="E3:21_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:21_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:21_1_1"/><b>then </b>
<font color="Maroon" title="c1">R</font> is <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:21_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="lfuzzy_0.html#T6" target="_self" title="LFUZZY_0:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E7:21_1_1"/>
<font color="Maroon" title="c1">R</font> is <a href="yellow_0.html#V3" title="YELLOW_0:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E4:21_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="yellow_0.html#D6" title="YELLOW_0:def.6">YELLOW_0:def 6</a></i>;<br/>


</div>
