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<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V1" target="_self">lower</a> <a href="waybel_9.html#NM1">TopLattice</a>, <font color="Maroon">T2</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V1" target="_self">lower</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green">E30</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T2</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span> #)
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a><br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T2</font>,<font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a>  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:26"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a><br/>

<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:26_1"/><i><font color="Green">E36</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">T2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<a NAME="E3:26_1"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span> &amp;  <a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">T</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a>, <a class="ref" href="yellow_0.html#T14">YELLOW_0:14</a></i>;<br/>
<b>thus </b><a NAME="E4:26_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<b>thus </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="waybel_2.html#K4">inf_op</a> <font color="Maroon">T</font></span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="yellow_0.html#T27">YELLOW_0:27</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i/>;<br/>


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