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<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon" title="c2">L</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E1:73"/><i><font color="Green" title="E32">A1</font></i>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</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="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1" title="YELLOW_1:def.1">YELLOW_1:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:73"/><i><font color="Green" title="E33">A2</font></i>: 
( <font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR3" title="COMPTS_1:NR.3">is_T3</a>  &amp;  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> )
 ;<br/>

<a NAME="E3:73"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
;<br/>
<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D9" target="_self" title="WAYBEL_3:def.9">WAYBEL_3:def 9</a><br/><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="c1">T</font>;<br/>



<b>assume </b><a NAME="E4:73"/><i><font color="Green" title="E35">A4</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c4">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">X</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="txt" href="waybel_3.html#E1:73"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<font color="Maroon" title="c5">a</font> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c5">a</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_3.html#E3:73"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="waybel_3.html#D5" target="_self" title="WAYBEL_3:def.5">Def5</a></i>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c5">a</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c6">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:73"/><i><font color="Green" title="E36">A5</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E4:73"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E9:73"/>
<font color="Maroon" title="c6">Y</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E8:73"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">Y</font> = <font color="Maroon" title="c6">Y</font> as   <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="txt" href="waybel_3.html#E1:73"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span><b> such that </b><br/><a NAME="E12:73"/><i><font color="Green" title="E37">A6</font></i>: 
( <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c8">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E8:73"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/>
<a NAME="E13:73"/>
<font color="Maroon" title="c7">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E1:73"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="waybel_3.html#E8:73"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <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><br/><a NAME="E15:73"/><i><font color="Green" title="E38">A7</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Y</font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:73"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="waybel_3.html#E8:73"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="urysohn1.html#T29" title="URYSOHN1:th.29">URYSOHN1:29</a></i>;<br/>
<b>take </b><font color="Maroon" title="c10">Z</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">W</font>;<br/>

<a NAME="E16:73"/>
<font color="Maroon" title="c9">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E17:73"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c10">Z</font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E15:73"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="tops_1.html#T54" title="TOPS_1:th.54">TOPS_1:54</a></i>;<br/>

<a NAME="E18:73"/>
<font color="Maroon" title="c8">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E12:73"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="waybel_3.html#T1" target="_self" title="WAYBEL_3:th.1">Th1</a></i>;<br/>
<a NAME="E19:73"/><b>then </b>
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E12:73"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E20:73"/>
<font color="Maroon" title="c10">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E15:73"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a><br/>

<b>assume </b><a NAME="E21:73"/><i><font color="Green" title="E39">A8</font></i>: 
( <font color="Maroon" title="c11">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c11">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c12">ZZ</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">ZZ</font> = <font color="Maroon" title="c12">ZZ</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">FZ</font> = <font color="Maroon" title="c11">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c13">ZZ</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E25:73"/>
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>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">ZZ</font> holds <br/><font color="Olive" title="b1">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E26:73"/><b>then </b>
<font color="Maroon" title="c13">ZZ</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E27:73"/><b>then </b>
<font color="Maroon" title="c14">FZ</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E21:73"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="tops_2.html#T20" title="TOPS_2:th.20">TOPS_2:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">FF</font> = <font color="Maroon" title="c14">FZ</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T25" title="YELLOW_1:th.25">YELLOW_1:25</a></i>;<br/>
<i><font color="Green" title="E40">A9</font></i>:  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c14">FZ</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c13">ZZ</font></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>
;<br/>
<a NAME="E30:73"/>
<font color="Maroon" title="c10">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">F</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E21:73"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>;<br/>
<a NAME="E31:73"/><b>then </b>
<font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E29:73"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E32:73"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E33:73"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font>
 
;<br/>
<a NAME="E34:73"/><b>then </b>
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E35:73"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c15">FF</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span><b> such that </b><br/><a NAME="E37:73"/><i><font color="Green" title="E41">A10</font></i>: 
( <font color="Maroon" title="c16">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">FF</font> &amp; <font color="Maroon" title="c8">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c16">A</font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E12:73"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="waybel_3.html#T18" target="_self" title="WAYBEL_3:th.18">Th18</a></i>;<br/>
<a NAME="E38:73"/><i><font color="Green" title="E42">A11</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c16">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c16">A</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></i>;<br/>
<a NAME="E39:73"/>
<font color="Maroon" title="c16">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">ZZ</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E1:73"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">G</font> = <font color="Maroon" title="c16">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c13">ZZ</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c17">G</font>
;<br/>

<b>thus </b><a NAME="E41:73"/>
<font color="Maroon" title="c17">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F</font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E37:73"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="xboole_1.html#T43" title="XBOOLE_1:th.43">XBOOLE_1:43</a></i>;<br/>

<b>thus </b><a NAME="E42:73"/>
<font color="Maroon" title="c10">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c17">G</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:73_3"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font>
 ;<br/>

<a NAME="E2:73_3"/><b>then </b><i><font color="Green" title="E43">A12</font></i>: 
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c16">A</font> &amp; not <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="txt" href="waybel_3.html#E12:73"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="waybel_3.html#E15:73"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="waybel_3.html#E37:73"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="waybel_3.html#E38:73"><i><font color="Green" title="E42">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:73_3"/><i><font color="Green" title="E44">A13</font></i>: 
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B</font> &amp; <font color="Maroon" title="c19">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:73_3"/>
not <font color="Maroon" title="c19">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">ZZ</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E2:73_3"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="waybel_3.html#E4:73_3"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:73_3"/><b>then </b>
<font color="Maroon" title="c19">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E4:73_3"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:73_3"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c17">G</font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E4:73_3"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E43:73"/>
<font color="Maroon" title="c17">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>


</div>
