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<b>let </b><font color="Maroon" title="c1">TS</font> be   <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">GX</font> being   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> <br/> for <font color="Olive" title="b2">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><br/> for <font color="Olive" title="b3">R</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">GX</font> holds <br/> ( ( <font color="Olive" title="b3">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b3">R</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> implies <font color="Olive" title="b3">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> ) &amp; ( <font color="Olive" title="b2">P</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> implies ( <font color="Olive" title="b2">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">P</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">GX</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</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">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><br/> for <font color="Olive" title="b2">R</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">GX</font> holds <br/> ( ( <font color="Olive" title="b2">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">R</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> implies <font color="Olive" title="b2">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> ) &amp; ( <font color="Olive" title="b1">P</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> implies ( <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b1">P</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> ) ) )</span><br/><b>let </b><font color="Maroon" title="c3">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>; <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">R</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">GX</font> holds <br/> ( ( <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b1">R</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> implies <font color="Olive" title="b1">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> ) &amp; ( <font color="Maroon" title="c3">P</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> implies ( <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> ) ) )</span><br/><b>let </b><font color="Maroon" title="c4">R</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">GX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c4">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c4">R</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> implies <font color="Maroon" title="c4">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> ) &amp; ( <font color="Maroon" title="c3">P</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> implies ( <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</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="c3">P</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> implies ( <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> ) )</span>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:103_1"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c4">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E2:103_1"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c4">R</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a></span><br/><a NAME="E3:103_1"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c4">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="c4">R</font>
 <b>by </b><i><a class="txt" href="tops_1.html#E1:103_1"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/><a NAME="E4:103_1"/><i><font color="Green" title="E62">A4</font></i>: 
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">R</font> &amp; <font color="Maroon" title="c4">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">R</font></span>)</span> )
 <b>by </b><i><a class="txt" href="tops_1.html#E2:103_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="tops_1.html#D6" target="_self" title="TOPS_1:def.6">Def6</a></i>;<br/><a NAME="E5:103_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">R</font>
 <b>by </b><i><a class="txt" href="tops_1.html#E2:103_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="tops_1.html#E3:103_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="tops_1.html#D6" target="_self" title="TOPS_1:def.6">Def6</a></i>;<br/><a NAME="E6:103_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">R</font>
 <b>by </b><i><a class="txt" href="tops_1.html#E3:103_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/><a NAME="E7:103_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">R</font>
 <b>by </b><i><a class="txt" href="tops_1.html#E4:103_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E8:103_1"/>
<font color="Maroon" title="c4">R</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D7" target="_self" title="TOPS_1: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>

<b>assume </b><a NAME="E2:103"/><i><font color="Green" title="E59">A5</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> )</span><br/>

<a NAME="E3:103"/><b>then </b><i><font color="Green" title="E60">A6</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#D7" target="_self" title="TOPS_1:def.7">Def7</a></i>;<br/>
<a NAME="E4:103"/>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="tops_1.html#E80"><i><font color="Green" title="E33">Lm4</font></i></a></i>;<br/>
<b>then </b> <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c3">P</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tops_1.html#E2:103"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="tops_1.html#T102" target="_self" title="TOPS_1:th.102">Th102</a></i>
<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">P</font></span>)</span>
<b>by </b><i><a class="txt" href="tops_1.html#E2:103"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="tops_1.html#D7" target="_self" title="TOPS_1:def.7">Def7</a></i>
;<br/>
<a NAME="E6:103"/><b>then </b>
<font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T77" target="_self" title="TOPS_1:th.77">Th77</a></i>;<br/>
<a NAME="E7:103"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E8:103"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" target="_self" title="TOPS_1:th.44">Th44</a></i>;<br/>
<b>hence </b><a NAME="E9:103"/>
( <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> )
 <b>by </b><i><a class="txt" href="tops_1.html#E3:103"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="tops_1.html#D6" target="_self" title="TOPS_1:def.6">Def6</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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