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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_1"/>
( <font color="Maroon" title="c1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c2">H'</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c3">H</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T80" title="ZFMISC_1:th.80">ZFMISC_1:80</a></i>;<br/><a NAME="E3:25_1_1_1_1"/>
<font color="Maroon" title="c3">H</font> is <a href="taxonom2.html#V3" title="TAXONOM2:attr.3">hierarchic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">TAXONOM2:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1_1_1_1_1"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font> )
 ;<br/>

<a NAME="E2:25_1_1_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> )
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:25_1_1_1_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_1_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c5">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">y</font> )
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c4">H</font> = <font color="Maroon" title="c3">H</font> as    <a href="taxonom2.html#M1" title="TAXONOM2:mode.1">Hierarchy</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="ref" href="taxonom2.html#D4" target="_self" title="TAXONOM2:def.4">Def4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">H</font>
;<br/><a NAME="E5:25_1_1_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/><b>hence </b><a NAME="E6:25_1_1_1_1"/>
<font color="Maroon" title="c4">H</font> is <a href="abian.html#V2" title="ABIAN:attr.2">covering</a>
 <b>by </b><i><a class="ref" href="abian.html#T4" title="ABIAN:th.4">ABIAN:4</a></i>;<br/><b>thus </b><a NAME="E7:25_1_1_1_1"/>
<font color="Maroon" title="c4">H</font> is <a href="taxonom2.html#V5" title="TAXONOM2:attr.5">T_3</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom2.html#D6" target="_self" title="TAXONOM2:def.6">TAXONOM2:def 6</a><br/>

<b>assume </b><a NAME="E1:25_1_1_1_1_2"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E2:25_1_1_1_1_2"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="txt" href="taxonom2.html#E1:25_1_1_1_1_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<b>assume </b><a NAME="E3:25_1_1_1_1_2"/><i><font color="Green" title="E19">A5</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ;<br/>

<b>thus </b><a NAME="E4:25_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c5">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">B</font> )
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:25_1_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="taxonom2.html#E2:25_1_1_1_1_2"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="taxonom2.html#E3:25_1_1_1_1_2"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_1_2"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c2">H'</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><b>reconsider </b><font color="Maroon" title="c3">H</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">H1</font> = <font color="Maroon" title="c3">H</font> as    <a href="taxonom2.html#M1" title="TAXONOM2:mode.1">Hierarchy</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="ref" href="taxonom2.html#D4" target="_self" title="TAXONOM2:def.4">Def4</a>, <a class="ref" href="taxonom2.html#T6" target="_self" title="TAXONOM2:th.6">Th6</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">H1</font>
;<br/><b>thus </b><a NAME="E4:25_1_1_1_2"/>
<font color="Maroon" title="c4">H1</font> is <a href="abian.html#V2" title="ABIAN:attr.2">covering</a>
 <b>by </b><i><a class="txt" href="taxonom2.html#E1:25_1_1_1_2"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="abian.html#T4" title="ABIAN:th.4">ABIAN:4</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/><b>thus </b><a NAME="E5:25_1_1_1_2"/>
<font color="Maroon" title="c4">H1</font> is <a href="taxonom2.html#V5" title="TAXONOM2:attr.5">T_3</a>
 <b>by </b><i><a class="ref" href="taxonom2.html#T11" target="_self" title="TAXONOM2:th.11">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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