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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">R</font> is <a href="lfuzzy_0.html#V1" target="_self">real</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_0.html#D2" target="_self">LFUZZY_0:def 2</a><br/>

<b>given </b><font color="Maroon">x</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">y</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><a NAME="E2:23_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/>

<a NAME="E3:23_1_1"/>
 ex <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; ( for <font color="Olive">y</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>

<b>let </b><font color="Maroon">z</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="E2:23_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<b>hence </b><a NAME="E3:23_1_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:23_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">R</font> is <a href="yellow_0.html#V2">upper-bounded</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:23_1_1"/>
 ex <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; ( for <font color="Olive">y</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:23_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>

<b>let </b><font color="Maroon">z</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="E2:23_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<b>hence </b><a NAME="E3:23_1_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:23_1_1"/><b>then </b>
<font color="Maroon">R</font> is <a href="yellow_0.html#V1">lower-bounded</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:23_1_1"/>
<font color="Maroon">R</font> is <a href="yellow_0.html#V3">bounded</a>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D6">YELLOW_0:def 6</a></i>;<br/>


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