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<span class="kw">set </span><font color="Maroon" title="c2">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>,<span class="p1">(<span class="default"><a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c1">ET</font></span>)</span> #);<br/>
<a NAME="E1:46_1_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>,<span class="p1">(<span class="default"><a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c1">ET</font></span>)</span> #)  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>,<span class="p1">(<span class="default"><a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c1">ET</font></span>)</span> #) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>,<span class="p1">(<span class="default"><a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c1">ET</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#T14" target="_self" onmouseover="rs('fintopo7/T14')" onmouseout="rh()">Th9</a></span>;<br/>
<a NAME="E2:46_1_1"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">ET</font>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>,<span class="p1">(<span class="default"><a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c1">ET</font></span>)</span> #) &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>,<span class="p1">(<span class="default"><a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c1">ET</font></span>)</span> #) holds <br/><font color="Olive" title="b1">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c1">ET</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo7.html#T14" target="_self" onmouseover="rs('fintopo7/T14')" onmouseout="rh()">Th9</a></span>;<br/>
<a NAME="E3:46_1_1"/><span class="kw">then </span>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ET</font>,<span class="p1">(<span class="default"><a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c1">ET</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="#E1:46_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="fintopo7.html#T14" target="_self" onmouseover="rs('fintopo7/T14')" onmouseout="rh()">Th9</a>, <a class="ref" href="pre_topc.html#D1" onmouseover="rs('pre_topc/D1')" onmouseout="rh()">PRE_TOPC:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:46_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <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="c1">ET</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c1">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 <font color="Olive">b<sub>1</sub></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"> verum</span><br/>


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