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<b>thus </b><a NAME="E1:47"/>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> 
 <b>by </b><i><a class="txt" href="topgen_3.html#E18"><i><font color="Green" title="E15">Lm5</font></i></a>, <a class="txt" href="topgen_3.html#E20"><i><font color="Green" title="E16">Lm6</font></i></a>, <a class="ref" href="topgen_3.html#T20" target="_self" title="TOPGEN_3:th.20">Th20</a>, <a class="ref" href="waybel23.html#T74" title="WAYBEL23:th.74">WAYBEL23:74</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>assume </b><a NAME="E2:47"/>
not  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> 
 ;<br/>

<a NAME="E3:47"/><b>then </b><i><font color="Green" title="E36">A1</font></i>: 
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">B</font> being    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of  <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> <b> such that </b><br/><a NAME="E5:47"/><i><font color="Green" title="E37">A2</font></i>: 
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <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="c2">B</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T75" title="WAYBEL23:th.75">WAYBEL23:75</a></i>;<br/>
<a NAME="E6:47"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="topgen_3.html#D2" target="_self" title="TOPGEN_3:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">x</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> , <font color="Maroon" title="c4">q</font> being  <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:47"/><i><font color="Green" title="E38">A3</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> &amp; not <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">q</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:47"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="topgen_3.html#E5:47"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="topgen_3.html#T32" target="_self" title="TOPGEN_3:th.32">Th32</a></i>;<br/>
<a NAME="E9:47"/>
<span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">q</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> 
 
<b>by </b><i><a class="ref" href="topgen_3.html#T11" target="_self" title="TOPGEN_3:th.11">Th11</a></i>;<br/>
<a NAME="E10:47"/><b>then </b>
<span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">q</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:47"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_3.html#E8:47"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="yellow_9.html#T22" title="YELLOW_9:th.22">YELLOW_9:22</a></i>;<br/>


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