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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">FX</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/>



<b>assume </b><a NAME="E1:3"/>
<font color="Maroon" title="c2">FX</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font>
 ;<br/>

<a NAME="E2:3"/><b>then </b><i><font color="Green" title="E4">A1</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">FX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a></i>;<br/>
<b>thus </b><a NAME="E3:3"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">W</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<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>;<br/>

<a NAME="E1:3_1"/><i><font color="Green" title="E5">A2</font></i>: 
<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> <font color="Maroon" title="c2">FX</font>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E2:3"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E2:3_1"/>
 ex <font color="Olive" title="b1">W</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="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c4">W</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:3_1_1"/><i><font color="Green" title="E6">A3</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">W</font>
 <b>and </b><br/><a NAME="E3:3_1_1"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:3_1"><i><font color="Green" title="E5">A2</font></i></a>, <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">W</font> = <font color="Maroon" title="c4">W</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="pcomps_1.html#E3:3_1_1"><i><font color="Green" title="E7">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">W</font>
;<br/>

<b>thus </b><a NAME="E5:3_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font> &amp; <font color="Maroon" title="c5">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FX</font> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E2:3_1_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="pcomps_1.html#E3:3_1_1"><i><font color="Green" title="E7">A4</font></i></a></i>;<br/>


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


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


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