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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E16</font></i>: 
( not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:15_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:15_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel15.html#E3:15_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:15"><i><font color="Green">E14</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/><a NAME="E7:15_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T72">WAYBEL_0:72</a></i>;<br/><a NAME="E8:15_1"/><b>then </b><i><font color="Green">E19</font></i>: 
( not <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i><a class="txt" href="waybel15.html#E1:15_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel15.html#E5:15_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green">E20</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:15_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="waybel15.html#E1:15_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/><a NAME="E3:15_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel15.html#E1:15_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/><a NAME="E4:15_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel15.html#E2:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel15.html#E8:15_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/><a NAME="E5:15_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel15.html#E3:15_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/><b>hence </b><a NAME="E6:15_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/><a NAME="E7:15_1_1"/>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><a NAME="E8:15_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>thus </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel15.html#E3:15_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="waybel15.html#E5:15_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel15.html#E8:15_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:15_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15"/>
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>


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