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<span class="kw">reconsider </span><font color="Maroon" title="c1">C</font> =  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ;<br/>
<a NAME="E2:71_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">r</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T8" title="RELSET_1:th.8">RELSET_1:8</a></span>;<br/>
<a NAME="E4:71_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T30" title="ZFMISC_1:th.30">ZFMISC_1:30</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">A</font> =  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c1">C</font> #) as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="waybel_9.html#E32"><span class="lab"><font color="Green" title="E25">Lm4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">A</font> as   <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> is  <a href="waybel_9.html#V1" title="WAYBEL_9:attr.1">strict</a>  &amp; not <font color="Maroon" title="c4">A</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E7:71_1_1"/>
( <font color="Maroon" title="c4">A</font> is  <a href="waybel_9.html#V1" title="WAYBEL_9:attr.1">strict</a>  &amp; not <font color="Maroon" title="c4">A</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  &amp; <font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E8:71_1_1"/>
<font color="Maroon" title="c4">A</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">A</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pre_topc.html#D16" title="PRE_TOPC:def.16">PRE_TOPC:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> or  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A</font>) st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E9:71_1_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
not <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A</font>) st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<a NAME="E10:71_1_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:71_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A</font>) st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive">b<sub>2</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive">b<sub>2</sub></font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:71_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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