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<span class="kw">thus </span><a NAME="E1:45"/>
 <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="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="topgen_3.html#E18"><span class="lab"><font color="Green" title="E15">Lm5</font></span></a>, <a class="txt" href="topgen_3.html#E20"><span class="lab"><font color="Green" title="E16">Lm6</font></span></a>, <a class="ref" href="topgen_3.html#T20" target="_self" title="TOPGEN_3:th.20">Th20</a>, <a class="ref" href="waybel23.html#T73" title="WAYBEL23:th.73">WAYBEL23:73</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">B</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a><span class="kw"> such that </span><br/><a NAME="E3:45"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#T74" title="WAYBEL23:th.74">WAYBEL23:74</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:45"/>
 not  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 <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#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T4" title="CARD_1:th.4">CARD_1:4</a></span>;<br/>
<a NAME="E6:45"/>
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_3.html#D2" target="_self" title="TOPGEN_3:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E8:45"/>
<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>
 <span class="kw">and </span><br/><a NAME="E9:45"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 not <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">q</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="topgen_3.html#T32" target="_self" title="TOPGEN_3:th.32">Th32</a></span>;<br/>
<a NAME="E10:45"/>
<span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">q</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_3.html#T11" target="_self" title="TOPGEN_3:th.11">Th11</a></span>;<br/>
<a NAME="E11:45"/><span class="kw">then </span>
<span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">q</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:45"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="yellow_9.html#T22" title="YELLOW_9:th.22">YELLOW_9:22</a></span>; <a class="txt" 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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