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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> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">b</font></span>)</span> where <font color="Olive" title="b1">b</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span> ;<br/>
<a NAME="E1:115"/><i><font color="Green" title="E65">A1</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
;<br/>

<a NAME="E1:115_1"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T2" title="CANTOR_1:th.2">CANTOR_1:2</a></i>;<br/>
<b>hence </b><a NAME="E2:115_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:115"/><i><font color="Green" title="E66">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 <b>and </b><br/><a NAME="E4:115"/><i><font color="Green" title="E67">A3</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/><font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="waybel23.html#E1:115"><i><font color="Green" title="E65">A1</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c3">b</font> being    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:115"/><i><font color="Green" title="E68">A4</font></i>: 
<font color="Maroon" title="c2">A</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">b</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E3:115"><i><font color="Green" title="E66">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">b</font>
;<br/>

<b>set </b><font color="Maroon" title="c4">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">b1</font></span>)</span> where <font color="Olive" title="b1">b1</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span> ;<br/>
<a NAME="E7:115"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T2" title="CANTOR_1:th.2">CANTOR_1:2</a></i>;<br/>
<a NAME="E8:115"/><b>then </b><i><font color="Green" title="E69">A5</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">b1</font></span>)</span> where <font color="Olive" title="b1">b1</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:115_2"/>
( ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">b1</font></span>)</span> where <font color="Olive" title="b2">b1</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span>  holds <br/><font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">y</font> ) implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="waybel23.html#E3:115"><i><font color="Green" title="E66">A2</font></i></a></i>;<br/><b>assume </b><a NAME="E2:115_2"/><i><font color="Green" title="E70">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:115_2"/><i><font color="Green" title="E71">A7</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">b1</font></span>)</span> where <font color="Olive" title="b1">b1</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c7">B2</font> being    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:115_2"/><i><font color="Green" title="E72">A8</font></i>: 
<font color="Maroon" title="c6">y</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="c7">B2</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c6">y</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="waybel23.html#E5:115_2"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/><a NAME="E7:115_2"/>
<font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">y1</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E4:115"><i><font color="Green" title="E67">A3</font></i></a>, <a class="txt" href="waybel23.html#E3:115_2"><i><font color="Green" title="E71">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E8:115_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E2:115_2"><i><font color="Green" title="E70">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:115"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E6:115"><i><font color="Green" title="E68">A4</font></i></a>, <a class="txt" href="waybel23.html#E8:115"><i><font color="Green" title="E69">A5</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


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