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<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>let </b><font color="Maroon" title="c2">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>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p3">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_1"/><i><font color="Green" title="E24">A1</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">A</font>
 <b>and </b><br/><a NAME="E4:43_1"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">A</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="tdlat_2.html#E4:43_1"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">B</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><br/><a NAME="E7:43_1"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">B</font>
 <b>and </b><br/><a NAME="E8:43_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:43_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="tdlat_2.html#D1" target="_self" title="TDLAT_2:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">D</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><br/><a NAME="E10:43_1"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">D</font>
 <b>and </b><br/><a NAME="E11:43_1"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E8:43_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">E</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><br/><a NAME="E13:43_1"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c8">E</font>
 <b>and </b><br/><a NAME="E14:43_1"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E11:43_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="tdlat_2.html#D1" target="_self" title="TDLAT_2:def.1">Def1</a></i>;<br/><a NAME="E15:43_1"/>
 ex <font color="Olive" title="b1">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">E</font></span>)</span>
;<br/>

<a NAME="E1:43_1_1"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E7:43_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="tdlat_2.html#E10:43_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="tdlat_2.html#E13:43_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="tdlat_2.html#T1" target="_self" title="TDLAT_2:th.1">Th1</a></i>;<br/>
<a NAME="E2:43_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E14:43_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1"/>
( <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> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">E</font></span>)</span> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">E</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span> )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E3:43_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="tdlat_2.html#E1:43_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="tdlat_2.html#D1" target="_self" title="TDLAT_2:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E16:43_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:43"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p3">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


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