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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</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">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a> iff <font color="Olive" title="b1">A</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a> iff <font color="Maroon" title="c2">A</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> implies <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a></span><br/><a NAME="E2:79_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#D7" target="_self" title="TEX_2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E3:79_1"/>
<font color="Maroon" title="c2">A</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T44" target="_self" title="TEX_2:th.44">Th44</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>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span>
<div class="add"><b>assume </b><a NAME="E1:79_2"/>
<font color="Maroon" title="c2">A</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a></span><br/><a NAME="E2:79_2"/><b>then </b><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T44" target="_self" title="TEX_2:th.44">Th44</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">D</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">discrete</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font> implies <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> )</span><br/><b>assume </b><a NAME="E1:79_2_1"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c3">D</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">discrete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font> implies <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> )</span><br/><b>assume </b><a NAME="E2:79_2_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font></span><br/><b>then reconsider </b><font color="Maroon" title="c4">E</font> = <font color="Maroon" title="c3">D</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E4:79_2_1"/>
<font color="Maroon" title="c4">E</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="#E1:79_2_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="tex_2.html#T44" target="_self" title="TEX_2:th.44">Th44</a></i>;<br/><b>hence </b><a NAME="E5:79_2_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font>
 <b>by </b><i><a class="txt" href="#E2:79_2_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="tex_2.html#T1" target="_self" title="TEX_2:th.1">Th1</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><b>hence </b><a NAME="E4:79_2"/>
<font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">maximal_discrete</a>
 <b>by </b><i><a class="txt" href="#E2:79_2"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tex_2.html#D7" target="_self" title="TEX_2:def.7">Def7</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>


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