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<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font>  st  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> &amp;  ex <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b2">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) holds <br/><font color="Olive" title="b1">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) implies <font color="Maroon" title="c2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> )</span><br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E37">A1</font></i>: 
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  holds <br/>(  not <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> or  not <font color="Olive" title="b1">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) or <font color="Maroon" title="c2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> )</span><br/>

<b>given </b><font color="Maroon" title="c3">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><a NAME="E3:59"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E4:59"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c3">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a></span><br/>

<a NAME="E5:59"/>
<font color="Maroon" title="c2">S</font> is <a href="waybel_0.html#V3" title="WAYBEL_0:attr.3">filtered-infs-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">Y</font> be  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D3" title="WAYBEL_0:def.3">WAYBEL_0:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">T</font> or  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )</span><br/>

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">T</font> or  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">Y</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_2.html#T7" title="YELLOW_2:th.7">YELLOW_2:7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">N</font> = <font color="Maroon" title="c5">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> ;<br/>
<b>assume </b><a NAME="E3:59_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E4:59_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="waybel19.html#T27" title="WAYBEL19:th.27">WAYBEL19:27</a></i>;<br/>
<a NAME="E5:59_1"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<a NAME="E6:59_1"/>
 <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c5">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel19.html#T43" title="WAYBEL19:th.43">WAYBEL19:43</a></i>;<br/>
<a NAME="E7:59_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c5">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="#E3:59"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="#E5:59_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="waybel21.html#T27" target="_self" title="WAYBEL21:th.27">Th27</a>, <a class="ref" href="waybel19.html#T26" title="WAYBEL19:th.26">WAYBEL19:26</a></i>;<br/>
<a NAME="E8:59_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c5">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="#E4:59"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<b>hence </b><a NAME="E9:59_1"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E3:59"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:59"/>
<font color="Maroon" title="c2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>
 <b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="waybel21.html#T16" target="_self" title="WAYBEL21:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
