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<span class="kw">let </span><font color="Maroon" title="c1">GX</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">R</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  holds <br/>( <font color="Olive" title="b1">R</font> is  <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>  iff <font color="Olive" title="b1">R</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">R</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  implies ( <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>  iff <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:131"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c2">R</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E2:131"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c2">R</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>  iff <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>  )</span><br/>

<a NAME="E3:131"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></span>;<br/>
<a NAME="E4:131"/>
<font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="tops_1.html#T55" target="_self" title="TOPS_1:th.55">Th55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:131"/>
( <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>  iff <font color="Maroon" title="c2">R</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="tops_1.html#D7" target="_self" title="TOPS_1:def.7">Def7</a>, <a class="ref" href="tops_1.html#D8" target="_self" title="TOPS_1:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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