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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">T2</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T2</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> #)
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a><br/>

<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T2</font>,<font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">T</font>
 ;<br/>

<a NAME="E3:26"/>
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">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" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c4">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <b>by </b><i><a class="txt" href="waybel19.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:26_1"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">T2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a><br/>

<a NAME="E3:26_1"/><b>then </b><i><font color="Green" title="E20">A4</font></i>: 
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c5">Y</font>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> &amp;  <a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">T</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="waybel19.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</a>, <a class="ref" href="yellow_0.html#T14" title="YELLOW_0:th.14">YELLOW_0:14</a></i>;<br/>
<b>thus </b><a NAME="E4:26_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>

<b>thus </b> <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c5">Y</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel19.html#E2:26"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="waybel19.html#E3:26_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c4">X</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel19.html#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="waybel19.html#E2:26"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="waybel19.html#E2:26_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="yellow_0.html#T27" title="YELLOW_0:th.27">YELLOW_0:27</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="waybel19.html#T9" target="_self" title="WAYBEL19:th.9">Th9</a></i>;<br/>


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