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<span class="kw">let </span><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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"> ( <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a>  implies  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c2">T2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T1</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span>verum;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">B1</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T1</font><span class="kw"> such that </span><br/><a NAME="E2:6"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">B1</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">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#T74" onmouseover="rs('waybel23/T74')" onmouseout="rh()">WAYBEL23:74</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:6"/>
<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</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="c2">T2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font><span class="kw"> such that </span><br/><a NAME="E5:6"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#D1" onmouseover="rs('t_0topsp/D1')" onmouseout="rh()">T_0TOPSP:def 1</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font> = <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> $1;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[$1] );<br/>
<a NAME="E6:6"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">w</font>] </span> } </span>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">B1</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="borsuk_2.html#S1" onmouseover="rs('borsuk_2/S1')" onmouseout="rh()">BORSUK_2:sch 1</a>();<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c5">B2</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T2</font><span class="kw"> such that </span><br/><a NAME="E8:6"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c5">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">w</font>] </span> } </span> 
 <span class="kw">from</span> <span class="lab"><a class="ref" href="lmod_7.html#S5" onmouseover="rs('lmod_7/S5')" onmouseout="rh()">LMOD_7:sch 5</a>();<br/></span>
<a NAME="E9:6"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font>  st <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font>  st <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/> ex <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</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"> ( <font color="Maroon" title="c6">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">A</font> holds <br/> ex <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1"/>
<font color="Maroon" title="c6">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">A</font> holds <br/> ex <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )</span><br/>

<a NAME="E2:6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c6">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="topgrp_1.html#T26" onmouseover="rs('topgrp_1/T26')" onmouseout="rh()">TOPGRP_1:26</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</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"> ( <font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">A</font> implies  ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:6_1"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">A</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">  ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )</span><br/>

<a NAME="E4:6_1"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:6_1"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E7:6_1"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E8:6_1"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c6">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E6:6_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E7:6_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">a1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font><span class="kw"> such that </span><br/><a NAME="E11:6_1"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
( <font color="Maroon" title="c10">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">a1</font> )
 <span class="kw">and </span><br/><a NAME="E12:6_1"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
<font color="Maroon" title="c10">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E8:6_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c11">a</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">a1</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"> ( <font color="Maroon" title="c11">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c11">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )</span><br/>

<a NAME="E13:6_1"/>
<font color="Maroon" title="c11">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c6">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:6_1"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="relat_1.html#T123" onmouseover="rs('relat_1/T123')" onmouseout="rh()">RELAT_1:123</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:6_1"/>
( <font color="Maroon" title="c11">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c11">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E4:6_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E6:6_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E7:6_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E11:6_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="funct_1.html#T77" onmouseover="rs('funct_1/T77')" onmouseout="rh()">FUNCT_1:77</a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</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="E10:6"/><span class="lab"><font color="Green" title="E9">A14</font></span>: 
<font color="Maroon" title="c3">B1</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 <font color="Maroon" title="c1">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T64" onmouseover="rs('tops_2/T64')" onmouseout="rh()">TOPS_2:64</a></span>;<br/>
<a NAME="E11:6"/>
<font color="Maroon" title="c5">B2</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 <font color="Maroon" title="c2">T2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</font> or <font color="Maroon" title="c6">x</font> <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 <font color="Maroon" title="c2">T2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B2</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"> <font color="Maroon" title="c6">x</font> <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 <font color="Maroon" title="c2">T2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">w</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font><span class="kw"> such that </span><br/><a NAME="E3:6_2"/><span class="lab"><font color="Green" title="E10">A15</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">w</font>)
 <span class="kw">and </span><br/><a NAME="E4:6_2"/><span class="lab"><font color="Green" title="E11">A16</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a></span>;<br/>
<a NAME="E5:6_2"/>
<font color="Maroon" title="c7">w</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:6"><span class="lab"><font color="Green" title="E9">A14</font></span></a>, <a class="txt" href="#E4:6_2"><span class="lab"><font color="Green" title="E11">A16</font></span></a></span>;<br/>
<a NAME="E6:6_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">w</font>) is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="topgrp_1.html#T25" onmouseover="rs('topgrp_1/T25')" onmouseout="rh()">TOPGRP_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:6_2"/>
<font color="Maroon" title="c6">x</font> <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 <font color="Maroon" title="c2">T2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_2"><span class="lab"><font color="Green" title="E10">A15</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">B2</font> = <font color="Maroon" title="c5">B2</font> as   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c2">T2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="yellow_9.html#T32" onmouseover="rs('yellow_9/T32')" onmouseout="rh()">YELLOW_9:32</a></span>;<br/>
<a NAME="E13:6"/>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c2">T2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel23.html#T73" onmouseover="rs('waybel23/T73')" onmouseout="rh()">WAYBEL23:73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:6"/>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c2">T2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E8:6"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E6">A3</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>
