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<span class="kw">let </span><font color="Maroon" title="c1">T</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">T</font> holds <br/> ( ( <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Olive" title="b1">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  iff <font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  ) ) &amp; ( <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a>  &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Olive" title="b1">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  iff <font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</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">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  iff <font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  ) ) &amp; ( <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  iff <font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  ) ) )</span><br/>



<a NAME="E1:17"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:17"/>
( <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  iff <font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compts_1.html#T10" target="_self" title="COMPTS_1:th.10">Th10</a></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="c1">T</font> is  <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  iff <font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a> 
 <span class="kw">and </span><br/><a NAME="E4:17"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  iff <font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">T9</font> = <font color="Maroon" title="c1">T</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E8">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">P9</font> = <font color="Maroon" title="c2">P</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E8">A3</font></span></a></span>;<br/>
<a NAME="E7:17"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">P9</font></span>)</span> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  iff <font color="Maroon" title="c3">T9</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">P9</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compts_1.html#T10" target="_self" title="COMPTS_1:th.10">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:17"/>
( <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  implies <font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="compts_1.html#T11" target="_self" title="COMPTS_1:th.11">Th11</a></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="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E9:17"/>
<font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</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">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/>

<a NAME="E10:17"/><span class="kw">then </span>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P</font></span>)</span>  st <font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> holds <br/><font color="Olive" title="b1">Q</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:17"/>
<font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="compts_1.html#T11" target="_self" title="COMPTS_1:th.11">Th11</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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