<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><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="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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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><span class="kw">hereby </span> <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="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"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:132_1"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
<font color="Maroon" title="c4">R</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E2:132_1"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E3:132_1"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132_1"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></span>;<br/><a NAME="E4:132_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132_1"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="tops_1.html#D6" target="_self" title="TOPS_1:def.6">Def6</a></span>;<br/><a NAME="E5:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A4</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:132_1"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></span>;<br/><a NAME="E6:132_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132_1"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="tops_1.html#D6" target="_self" title="TOPS_1:def.6">Def6</a></span>;<br/><a NAME="E7:132_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:132_1"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:132_1"/>
<font color="Maroon" title="c4">R</font> is  <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#D7" target="_self" title="TOPS_1:def.7">Def7</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:132"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E3:132"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="tops_1.html#E78"><span class="lab"><font color="Green" title="E27">Lm2</font></span></a></span>;<br/>
<span class="kw">then </span> <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="tops_1.html#T102" target="_self" title="TOPS_1:th.102">Th102</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="tops_1.html#D7" target="_self" title="TOPS_1:def.7">Def7</a></span>
;<br/>
<a NAME="E5:132"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T77" target="_self" title="TOPS_1:th.77">Th77</a></span>;<br/>
<a NAME="E6:132"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></span>;<br/>
<a NAME="E7:132"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A6</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T44" target="_self" title="TOPS_1:th.44">Th44</a></span>;<br/>
<a NAME="E8:132"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="tops_1.html#D7" target="_self" title="TOPS_1:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:132"/>
( <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:132"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="tops_1.html#D6" target="_self" title="TOPS_1:def.6">Def6</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/>


</div>
