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<span class="kw">let </span><font color="Maroon" title="c2">Y0</font> be    <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">Y0</font> is  <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>  implies <font color="Maroon" title="c2">Y0</font> is  <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a>  )</span><br/>

<span class="kw">reconsider </span><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">Y0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="tex_2.html#E10"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:85_3_1"/>
 not <font color="Maroon" title="c2">Y0</font> is  <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">Y0</font> is  <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a> </span><br/>

<a NAME="E3:85_3_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_2.html#T8" target="_self" title="TEX_2:th.8">Th13</a></span>;<br/>
<a NAME="E4:85_3_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">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="c2">Y0</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_2.html#T43" target="_self" title="TEX_2:th.43">Th49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:85_3_1"/>
<font color="Maroon" title="c2">Y0</font> is  <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_2.html#D6" target="_self" title="TEX_2:def.6">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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