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<b>let </b><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">X0</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>;<br/>



<a NAME="E1:63"/><i><font color="Green" title="E48">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="topgen_3.html#K7" title="TOPGEN_3:func.7">-DiscreteTop</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="topgen_3.html#D8" target="_self" title="TOPGEN_3:def.8">Def8</a></i>;<br/>
<b>thus </b><a NAME="E2:63"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="topgen_3.html#K7" title="TOPGEN_3:func.7">-DiscreteTop</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c3">X0</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</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="E1:63_1"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="topgen_3.html#K7" title="TOPGEN_3:func.7">-DiscreteTop</a> <font color="Maroon" title="c2">X</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="topgen_3.html#K7" title="TOPGEN_3:func.7">-DiscreteTop</a> <font color="Maroon" title="c2">X</font></span>)</span> ;<br/>
<a NAME="E3:63_1"/>
( <font color="Maroon" title="c5">a</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; ( ( <font color="Maroon" title="c5">a</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> &amp; not <font color="Maroon" title="c2">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or  not <font color="Maroon" title="c5">a</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:63"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="topgen_3.html#E1:63_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="topgen_3.html#E60"><i><font color="Green" title="E46">Lm7</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:63_1"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X0</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:63"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="topgen_3.html#T43" target="_self" title="TOPGEN_3:th.43">Th43</a>, <a class="ref" href="tex_2.html#T5" title="TEX_2:th.5">TEX_2:5</a></i>;<br/>
<a NAME="E5:63_1"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c3">X0</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:63_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c3">X0</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c4">a</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="E3:63"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c3">X0</font></span>)</span>
 ;<br/>

<a NAME="E4:63"/><b>then </b><i><font color="Green" title="E49">A3</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c3">X0</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:63"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c4">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X0</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="topgen_3.html#K7" title="TOPGEN_3:func.7">-DiscreteTop</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="topgen_3.html#E1:63"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>assume </b><a NAME="E7:63"/><i><font color="Green" title="E50">A4</font></i>: 
not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="topgen_3.html#K7" title="TOPGEN_3:func.7">-DiscreteTop</a> <font color="Maroon" title="c2">X</font></span>)</span>
 ;<br/>

<a NAME="E8:63"/><b>then </b><i><font color="Green" title="E51">A5</font></i>: 
not <font color="Maroon" title="c5">a</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:63"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X0</font> <a href="topgen_3.html#K7" title="TOPGEN_3:func.7">-DiscreteTop</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E7:63"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:63"/><b>then </b><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="topgen_3.html#D8" target="_self" title="TOPGEN_3:def.8">Def8</a></i>;<br/>
<a NAME="E11:63"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c5">a</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E9:63"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="tex_2.html#T5" title="TEX_2:th.5">TEX_2:5</a></i>;<br/>
<a NAME="E12:63"/><b>then </b>
not <font color="Maroon" title="c2">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:63"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="topgen_3.html#E60"><i><font color="Green" title="E46">Lm7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:63"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_3.html#E4:63"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="topgen_3.html#E8:63"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="topgen_3.html#E10:63"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="topgen_3.html#E11:63"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="topgen_3.html#T43" target="_self" title="TOPGEN_3:th.43">Th43</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div>
