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<span class="kw">let </span><font color="Maroon" title="c1">TM</font> be   <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">iC</font> being   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> holds <br/> (  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">iC</font> iff  <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">iC</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">iC</font> be   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</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"> (  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> iff  <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font><span class="kw"> such that </span><br/><a NAME="E2:38"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c3">A</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 <span class="kw">and </span><br/><a NAME="E3:38"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">TM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_1.html#D12" onmouseover="rs('topgen_1/D12')" onmouseout="rh()">TOPGEN_1:def 12</a></span>;<br/>
<div><span class="kw">hereby </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"> (  <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> implies  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:38_1"/>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</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">  <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font></span><br/><a NAME="E2:38_1"/><span class="kw">then </span>
 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">TM</font>  st <font color="Olive" title="b1">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; (  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TM</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">B</font> holds <br/><font color="Olive" title="b2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">B</font> ) holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">F</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#T21" target="_self" onmouseover="rs('metrizts/T21')" onmouseout="rh()">Th21</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1"/>
 <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="metrizts.html#E23"><span class="lab"><font color="Green" title="E17">Lm6</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/></div>
<span class="kw">end;</span></div>

<a NAME="E5:38"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="metrizts.html#T17" target="_self" onmouseover="rs('metrizts/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:38"/>
 <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</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">  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font></span><br/>

<a NAME="E7:38"/><span class="kw">then </span>
<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c2">iC</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="card_2.html#T90" onmouseover="rs('card_2/T90')" onmouseout="rh()">CARD_2:90</a></span>;<br/>
<a NAME="E8:38"/><span class="kw">then </span>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="card_2.html#K2" title="CARD_2:func.2">*`</a> <font color="Maroon" title="c2">iC</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E27">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:38"/>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">TM</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">iC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="metrizts.html#E18"><span class="lab"><font color="Green" title="E12">Lm5</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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