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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> holds <br/><font color="Olive" title="b2">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Olive" title="b1">M</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a></span><br/><b>let </b><font color="Maroon" title="c2">M</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> holds <br/><font color="Olive" title="b1">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c2">M</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a></span><br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> implies <font color="Maroon" title="c3">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c2">M</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> )</span><br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</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">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c2">M</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a></span><br/>

<b>let </b><font color="Maroon" title="c4">Q</font> be   <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">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2" target="_self" title="TOPS_2:def.2">TOPS_2:def 2</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">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c2">M</font> implies <font color="Maroon" title="c4">Q</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<b>assume </b><a NAME="E2:40"/>
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="tops_2.html#K1" title="TOPS_2:func.1">|</a> <font color="Maroon" title="c2">M</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">Q</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<b>then consider </b><font color="Maroon" title="c5">R</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:40"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c5">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>and </b><br/><a NAME="E5:40"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c5">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font>
 <b>by </b><i><a class="ref" href="tops_2.html#D3" target="_self" title="TOPS_2:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">R</font> = <font color="Maroon" title="c5">R</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E7:40"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c6">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="tops_2.html#E1:40"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="tops_2.html#E4:40"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="tops_2.html#D2" target="_self" title="TOPS_2:def.2">Def2</a></i>;<br/>
<a NAME="E8:40"/>
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<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">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="tops_2.html#E5:40"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:40"/>
<font color="Maroon" title="c4">Q</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E7:40"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</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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