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<b>let </b><font color="Maroon" title="c3">H</font>, <font color="Maroon" title="c4">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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> 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/>( $1 <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="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">F</font> );<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:9_1_2"/><i><font color="Green" title="E4">A1</font></i>: 
for <font color="Olive" title="b1">Z</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Z</font>] )
 <b>and </b><br/><a NAME="E2:9_1_2"/><i><font color="Green" title="E5">A2</font></i>: 
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> holds <br/> ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] )
 ;<br/>

<a NAME="E3:9_1_2"/>
<font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_2_1_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">Z'</font> = <font color="Maroon" title="c5">Z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E3:9_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">Z</font>]
 <b>by </b><i><a class="txt" href="topgen_1.html#E1:9_1_2"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="topgen_1.html#E1:9_1_2_1_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E4:9_1_2_1_1"/>
<font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="topgen_1.html#E2:9_1_2"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:9_1_2_1"/><b>then </b><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c3">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_2_1_2"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">X'</font> = <font color="Maroon" title="c5">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/><a NAME="E3:9_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">X'</font>]
 <b>by </b><i><a class="txt" href="topgen_1.html#E2:9_1_2"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_1.html#E1:9_1_2_1_2"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E4:9_1_2_1_2"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="topgen_1.html#E1:9_1_2"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_2_1"/>
<font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="topgen_1.html#E2:9_1_2_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:9_1_2"/>
<font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font>
 ;<br/>


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