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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> holds   <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Olive" title="b2">G</font></span>)</span></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span></span><br/>





<b>thus </b><a NAME="E1:38"/>
 <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E20">A1</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#K3" title="TOPGEN_1:func.3">Der</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span></span><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="E4:38_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G</font> )
 <b>by </b><i><a class="txt" href="#E1:38_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="topgen_1.html#D6" target="_self" title="TOPGEN_1:def.6">Def6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1"/>
( <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> or <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> )
 </span><b>by </b><i><a class="txt" href="#E4:38_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span></span><br/><div class="add"><a NAME="E2:38_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E4:38_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="topgen_1.html#D6" target="_self" title="TOPGEN_1:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_2"/>
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span></span><br/><div class="add"><a NAME="E2:38_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="#E4:38_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="topgen_1.html#D6" target="_self" title="TOPGEN_1:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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

<a NAME="E2:38"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_1.html#T28" target="_self" title="TOPGEN_1:th.28">Th28</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:38"/>
 <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_1.html#T28" target="_self" title="TOPGEN_1:th.28">Th28</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:38"/>
<span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:38"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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