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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>









<a NAME="E1:29"/><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:29"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> iff ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">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">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:29_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K4">\\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_5.html#T13" target="_self">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel_5.html#E3:29_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><a NAME="E6:29_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:29_1_1"><i><font color="Green">E20</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:29_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a>  implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:29"><i><font color="Green">E18</font></i></a>, <a class="ref" href="waybel_5.html#T13" target="_self">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">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK7" target="_self">Infs</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:29_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_5.html#T13" target="_self">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel_5.html#E3:29_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><a NAME="E6:29_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:29_2"><i><font color="Green">E20</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E4:29"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK7" target="_self">Infs</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:29"><i><font color="Green">E18</font></i></a>, <a class="ref" href="waybel_5.html#T13" target="_self">Th13</a></i>;<br/>


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