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<a NAME="E1:92_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/>
<b>thus </b><a NAME="E2:92_1_1"/>
( <font color="Maroon">f</font> is <a href="waybel34.html#V5" target="_self">bottom-preserving</a> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1_1_1"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel34.html#D16" target="_self">WAYBEL34:def 16</a><br/>

<a NAME="E2:92_1_1_1"/><b>then </b>
(  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font>,<font color="Maroon">S</font> implies  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<b>hence </b><a NAME="E3:92_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T42">YELLOW_0:42</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E3:92_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E4:92_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font>,<font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a>,<a class="ref" href="waybel34.html#D16" target="_self">WAYBEL34:def 16</a><br/>

<b>thus </b><a NAME="E5:92_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T42">YELLOW_0:42</a></i>;<br/>

<b>thus </b><a NAME="E6:92_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


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