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<b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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"> ( not <font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</a> implies  not <font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V5" title="TEX_2:attr.5">maximal_discrete</a> )</span><br/>

<b>assume </b><a NAME="E1:113_2_1"/><i><font color="Green" title="E54">A3</font></i>: 
not <font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</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="c2">X0</font> is <a href="tex_2.html#V5" title="TEX_2:attr.5">maximal_discrete</a></span><br/>

<b>assume </b><a NAME="E2:113_2_1"/><i><font color="Green" title="E55">A4</font></i>: 
<font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V5" title="TEX_2:attr.5">maximal_discrete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c3">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="tex_2.html#E17"><i><font color="Green" title="E9">Lm1</font></i></a></i>;<br/>
<a NAME="E4:113_2_1"/>
<font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a>
 
<b>by </b><i><a class="txt" href="tex_2.html#E2:113_2_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="tex_2.html#D8" target="_self" title="TEX_2:def.8">Def8</a></i>;<br/>
<a NAME="E5:113_2_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">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/>
<a NAME="E6:113_2_1"/><b>then </b><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c2">X0</font> is <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>
 
<b>by </b><i><a class="ref" href="tex_2.html#T26" target="_self" title="TEX_2:th.26">Th26</a></i>;<br/>
<a NAME="E7:113_2_1"/>
not <font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 
<b>by </b><i><a class="txt" href="tex_2.html#E1:113_2_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="tex_2.html#T13" target="_self" title="TEX_2:th.13">Th13</a></i>;<br/>
<a NAME="E8:113_2_1"/><b>then </b>
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c1">X</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="tex_2.html#T5" target="_self" title="TEX_2:th.5">Th5</a>, <a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a></i>;<br/>
<a NAME="E9:113_2_1"/><b>then </b>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">X0</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> #)
 
<b>by </b><i><a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></i>;<br/>
<b>hence </b><a NAME="E10:113_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_2.html#E6:113_2_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="tex_2.html#T17" target="_self" title="TEX_2:th.17">Th17</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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