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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st (  for <font color="Olive" title="b2">B1</font>, <font color="Olive" title="b3">B2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">B</font>  ex <font color="Olive" title="b4">BB</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">B</font> st <font color="Olive" title="b2">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b4">BB</font> ) &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">B</font> holds <br/> ex <font color="Olive" title="b2">ET</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">ET</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">B</font> is   <a href="fintopo7.html#NM4" title="FINTOPO7:NM.4">Basis</a> of <font color="Olive" title="b2">ET</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font>  ex <font color="Olive" title="b3">BB</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">B</font> st <font color="Olive" title="b1">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b3">BB</font> ) &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">B</font> implies  ex <font color="Olive" title="b1">ET</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">ET</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">B</font> is   <a href="fintopo7.html#NM4" title="FINTOPO7:NM.4">Basis</a> of <font color="Olive" title="b1">ET</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">B2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">B</font>  ex <font color="Olive" title="b3">BB</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">B</font> st <font color="Olive" title="b1">B1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b3">BB</font>
 <span class="kw">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">ET</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">ET</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">B</font> is   <a href="fintopo7.html#NM4" title="FINTOPO7:NM.4">Basis</a> of <font color="Olive" title="b1">ET</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">O</font> =  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">B</font></span>)</span> #);<br/>
<a NAME="E3:43"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">B</font></span>)</span> #) is   <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="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="fintopo7.html#T4" target="_self" onmouseover="rs('fintopo7/T4')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">ET</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a><span class="kw"> such that </span><br/><a NAME="E5:43"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">B</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">ET</font>
 <span class="kw">and </span><br/><a NAME="E6:43"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c5">ET</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">B</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#T19" target="_self" onmouseover="rs('fintopo7/T19')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">B1</font> = <font color="Maroon" title="c2">B</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c5">ET</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<a NAME="E8:43"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c6">B1</font> is  <a href="fintopo7.html#V7" title="FINTOPO7:attr.7">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="cantor_1.html#T1" onmouseover="rs('cantor_1/T1')" onmouseout="rh()">CANTOR_1:1</a></span>;<br/>
<a NAME="E9:43"/>
<font color="Maroon" title="c6">B1</font> is  <a href="fintopo7.html#V6" title="FINTOPO7:attr.6">quasi_basis</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E26">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:43"/>
 ex <font color="Olive" title="b1">ET</font> being   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">ET</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">B</font> is   <a href="fintopo7.html#NM4" title="FINTOPO7:NM.4">Basis</a> of <font color="Olive" title="b1">ET</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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