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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K5" title="RUSUB_5:func.5">TopUnitSpace</a> <font color="Maroon" title="c1">V</font></span>)</span><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b3">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b2">r</font>) holds <br/><font color="Olive" title="b1">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K5" title="RUSUB_5:func.5">TopUnitSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b2">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">r</font>) holds <br/><font color="Maroon" title="c2">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> (<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c3">r</font>) holds <br/><font color="Maroon" title="c2">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>; <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">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> (<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c3">r</font>) implies <font color="Maroon" title="c2">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:56"/>
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> (<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c3">r</font>)
 ; <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">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:56"/><span class="kw">then </span>
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_5.html#T37" target="_self" title="RUSUB_5:th.37">Th37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56"/>
<font color="Maroon" title="c2">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</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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