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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">T'</font> = <font color="Maroon" title="c1">T</font> as  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_9.html#T44" title="YELLOW_9:th.44">YELLOW_9:44</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">dx</font> =  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T'</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T'</font> ;<br/>
<a NAME="E4:12"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">x</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T11" title="WAYBEL11:th.11">WAYBEL11:11</a></i>;<br/>
<a NAME="E5:12"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
;<br/>
<a NAME="E6:12"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E7:12"/><b>then </b><i><font color="Green" title="E8">A2</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></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="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T'</font>;<br/><b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">D</font> = <font color="Maroon" title="c6">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T'</font> ;<br/><b>assume </b><a NAME="E3:12_1"/>
<font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/><a NAME="E4:12_1"/><b>then </b><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c7">D</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>
 <b>by </b><i><a class="ref" href="waybel11.html#T7" title="WAYBEL11:th.7">WAYBEL11:7</a></i>;<br/><a NAME="E5:12_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="txt" href="waybel32.html#E1:12_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E6:12_1"/>
<font color="Maroon" title="c4">dx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="txt" href="waybel32.html#E4:12_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="waybel11.html#T6" title="WAYBEL11:th.6">WAYBEL11:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:12"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="waybel32.html#E4:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="waybel32.html#E7:12"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="yellow_8.html#T17" title="YELLOW_8:th.17">YELLOW_8:17</a></i>;<br/>


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