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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</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="E1:12"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font>
 ;<br/>

<a NAME="E2:12"/>
 <a href="topgen_1.html#K1" title="TOPGEN_1:func.1">Fr</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_1.html#K1" title="TOPGEN_1:func.1">Fr</a> <font color="Maroon" title="c3">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</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="E1:12_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_1.html#K1" title="TOPGEN_1:func.1">Fr</a> <font color="Maroon" title="c2">F</font>
 ;<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#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<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="E4:12_1"/><i><font color="Green" title="E8">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#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c6">B</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="topgen_1.html#E1:12_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="topgen_1.html#D1" target="_self" title="TOPGEN_1:def.1">Def1</a></i>;<br/>
<b>thus </b><a NAME="E5:12_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_1.html#K1" title="TOPGEN_1:func.1">Fr</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="topgen_1.html#E1:12"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="topgen_1.html#E4:12_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="topgen_1.html#D1" target="_self" title="TOPGEN_1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12"/>
 <a href="topgen_1.html#K1" title="TOPGEN_1:func.1">Fr</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_1.html#K1" title="TOPGEN_1:func.1">Fr</a> <font color="Maroon" title="c3">G</font>
 ;<br/>


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