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<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">J</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">K</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">J</font>;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">L</font>;<br/>









<a NAME="E1:29"/><i><font color="Green" title="E18">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:29"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span> iff  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font>
 <b>and </b><br/><a NAME="E4:29_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span>,<font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="waybel_5.html#T13" target="_self" title="WAYBEL_5:th.13">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c6">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> <b>by </b><i><a class="txt" href="waybel_5.html#E3:29_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c7">j</font>;<br/><b>thus </b><a NAME="E6:29_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:29_1_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:29_1"/>
(  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:29"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="waybel_5.html#T13" target="_self" title="WAYBEL_5:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:29_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK7" title="WAYBEL_5:NK.7">Infs</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_2"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font>
 <b>and </b><br/><a NAME="E4:29_2"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span>,<font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="waybel_5.html#T13" target="_self" title="WAYBEL_5:th.13">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c6">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> <b>by </b><i><a class="txt" href="waybel_5.html#E3:29_2"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c7">j</font>;<br/><b>thus </b><a NAME="E6:29_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:29_2"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E4:29"/>
(  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK7" title="WAYBEL_5:NK.7">Infs</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:29"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="waybel_5.html#T13" target="_self" title="WAYBEL_5:th.13">Th13</a></i>;<br/>


</div>
