<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</font> be   <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"> (  not <font color="Maroon" title="c1">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">T2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  implies (  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> &amp;  <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> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> ) )</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">set </span><font color="Maroon" title="c3">PR2</font> =  <a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">PR1</font> =  <a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>);<br/>
<span class="kw">assume </span><a NAME="E1:14"/>
(  not <font color="Maroon" title="c1">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">T2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">T1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> &amp;  <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> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">t1</font> = <font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c6">t2</font> = <font color="Maroon" title="c2">T2</font> as   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> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">T12</font> = <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">t1</font>,<font color="Maroon" title="c6">t2</font></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> as   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> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">B12</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c7">T12</font><span class="kw"> such that </span><br/><a NAME="E5:14"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">B12</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="c7">T12</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">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="c7">T12</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="c1">T1</font> = <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> <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="c8">B12</font> &amp; <font color="Maroon">S<sub>1</sub></font>[$1] );<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">B1</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T1</font><span class="kw"> such that </span><br/><a NAME="E7:14"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c9">B1</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="c7">T12</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="E8:14"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c8">B12</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="c7">T12</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="E9:14"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c9">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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B1</font> or <font color="Maroon" title="c10">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="c1">T1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B1</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="c10">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="c1">T1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">w</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T12</font><span class="kw"> such that </span><br/><a NAME="E3:14_1"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">w</font>)
 <span class="kw">and </span><br/><a NAME="E4:14_1"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>;<br/>
<a NAME="E5:14_1"/>
<font color="Maroon" title="c11">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="#E8:14"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E4:14_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a></span>;<br/>
<a NAME="E6:14_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">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="ref" href="borsuk_1.html#T18" onmouseover="rs('borsuk_1/T18')" onmouseout="rh()">BORSUK_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:14_1"/>
<font color="Maroon" title="c10">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="c1">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E12">A5</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="E10:14"/>
 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="c1">T1</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="c1">T1</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="c1">T1</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="c9">B1</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="14_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</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="c10">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="c1">T1</font>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">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="c1">T1</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="c9">B1</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="c10">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_2"/>
<font color="Maroon" title="c10">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="c1">T1</font>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">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="c1">T1</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="c9">B1</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="c10">A</font> )</span><br/>

<a NAME="E2:14_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">A</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T6" onmouseover="rs('borsuk_1/T6')" onmouseout="rh()">BORSUK_1:6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">p2</font> =  the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font>;<br/>
<a NAME="E3:14_2"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
 the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">t2</font>
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">p1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</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="c12">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">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="c1">T1</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="c9">B1</font> &amp; <font color="Maroon" title="c12">p1</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="c10">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:14_2"/>
<font color="Maroon" title="c12">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">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="c1">T1</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="c9">B1</font> &amp; <font color="Maroon" title="c12">p1</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="c10">A</font> )</span><br/>

<a NAME="E5:14_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">p1</font>, the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">A</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_2"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">p</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">p1</font>, the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">T12</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">a12</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T12</font><span class="kw"> such that </span><br/><a NAME="E8:14_2"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<font color="Maroon" title="c14">a12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">B12</font>
 <span class="kw">and </span><br/><a NAME="E9:14_2"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">a12</font>
 <span class="kw">and </span><br/><a NAME="E10:14_2"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
<font color="Maroon" title="c14">a12</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">A</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_2"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E5:14_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/>
<a NAME="E11:14_2"/>
( <font color="Maroon" title="c12">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">t1</font> &amp;  the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">t2</font> )
 
;<br/>
<a NAME="E12:14_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">p1</font>, the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E13:14_2"/>
<font color="Maroon" title="c14">a12</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="#E8:14"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E8:14_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">a</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c14">a12</font>) as   <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 <font color="Maroon" title="c1">T1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T18" onmouseover="rs('borsuk_1/T18')" onmouseout="rh()">BORSUK_1:18</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">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"> ( <font color="Maroon" title="c15">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">B1</font> &amp; <font color="Maroon" title="c12">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c15">a</font> &amp; <font color="Maroon" title="c15">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )</span><br/>

 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> = 
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">T12</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D2" onmouseover="rs('borsuk_1/D2')" onmouseout="rh()">BORSUK_1:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:14_2"/>
( <font color="Maroon" title="c15">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">B1</font> &amp; <font color="Maroon" title="c12">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c15">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E8:14_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E9:14_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E12:14_2"><span class="lab"><font color="Green" title="E18">A13</font></span></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"> <font color="Maroon" title="c15">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c16">y</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="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">a</font> or <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E17:14_2"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">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"> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E19:14_2"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
<font color="Maroon" title="c17">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> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E20:14_2"/><span class="lab"><font color="Green" title="E20">A15</font></span>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">a12</font> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">x1</font>, <font color="Maroon" title="c19">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E22:14_2"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
( <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> )
 <span class="kw">and </span><br/><a NAME="E23:14_2"/><span class="lab"><font color="Green" title="E22">A17</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:14_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E24:14_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:14_2"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:14_2"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14_2"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E20:14_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E23:14_2"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</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="c10">B1</font> = <font color="Maroon" title="c9">B1</font> as   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">T1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E11">A4</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="E12:14"/><span class="lab"><font color="Green" title="E12">A18</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="c7">T12</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="c8">B12</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>
<a NAME="E13:14"/>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c5">t1</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="c10">B1</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="E14:14"/>
 <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <font color="Maroon" title="c1">T1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="waybel23.html#K2" title="WAYBEL23:func.2">weight</a> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E12:14"><span class="lab"><font color="Green" title="E12">A18</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">  <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> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span></span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T12</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> = <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">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="E16:14"/><span class="lab"><font color="Green" title="E13">A19</font></span>: 
<font color="Maroon" title="c11">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>2</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="c7">T12</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="E17:14"/><span class="lab"><font color="Green" title="E14">A20</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="c11">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="14_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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="c12">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="c12">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="c11">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="c12">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_3"/>
<font color="Maroon" title="c12">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="c12">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="c11">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="c12">A</font> )</span><br/>

<a NAME="E2:14_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A21</font></span>: 
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<font color="Maroon" title="c12">A</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T6" onmouseover="rs('borsuk_1/T6')" onmouseout="rh()">BORSUK_1:6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">p1</font> =  the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</font>;<br/>
<a NAME="E3:14_3"/><span class="lab"><font color="Green" title="E16">A22</font></span>: 
 the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">t1</font>
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c14">p2</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="c14">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">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="c11">B2</font> &amp; <font color="Maroon" title="c14">p2</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="c12">A</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:14_3"/>
<font color="Maroon" title="c14">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">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="c11">B2</font> &amp; <font color="Maroon" title="c14">p2</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="c12">A</font> )</span><br/>

<a NAME="E5:14_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A23</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"> the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c14">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<font color="Maroon" title="c12">A</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_3"><span class="lab"><font color="Green" title="E16">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">p</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"> the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c14">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">T12</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">a12</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T12</font><span class="kw"> such that </span><br/><a NAME="E8:14_3"/><span class="lab"><font color="Green" title="E18">A24</font></span>: 
<font color="Maroon" title="c16">a12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">B12</font>
 <span class="kw">and </span><br/><a NAME="E9:14_3"/><span class="lab"><font color="Green" title="E19">A25</font></span>: 
<font color="Maroon" title="c15">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">a12</font>
 <span class="kw">and </span><br/><a NAME="E10:14_3"/><span class="lab"><font color="Green" title="E20">A26</font></span>: 
<font color="Maroon" title="c16">a12</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<font color="Maroon" title="c12">A</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_3"><span class="lab"><font color="Green" title="E15">A21</font></span></a>, <a class="txt" href="#E5:14_3"><span class="lab"><font color="Green" title="E17">A23</font></span></a>, <a class="ref" href="yellow_9.html#T31" onmouseover="rs('yellow_9/T31')" onmouseout="rh()">YELLOW_9:31</a></span>;<br/>
<a NAME="E11:14_3"/>
(  the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">t1</font> &amp; <font color="Maroon" title="c14">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">t2</font> )
 
;<br/>
<a NAME="E12:14_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A27</font></span>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> ( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c14">p2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" onmouseover="rs('funct_3/D5')" onmouseout="rh()">FUNCT_3:def 5</a></span>;<br/>
<a NAME="E13:14_3"/>
<font color="Maroon" title="c16">a12</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="#E8:14"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E8:14_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">a</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c16">a12</font>) as   <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 <font color="Maroon" title="c2">T2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T18" onmouseover="rs('borsuk_1/T18')" onmouseout="rh()">BORSUK_1:18</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">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"> ( <font color="Maroon" title="c17">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">B2</font> &amp; <font color="Maroon" title="c14">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">a</font> &amp; <font color="Maroon" title="c17">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font> )</span><br/>

 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> = 
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font></span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D5" onmouseover="rs('funct_3/D5')" onmouseout="rh()">FUNCT_3:def 5</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">T12</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D2" onmouseover="rs('borsuk_1/D2')" onmouseout="rh()">BORSUK_1:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:14_3"/>
( <font color="Maroon" title="c17">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">B2</font> &amp; <font color="Maroon" title="c14">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c17">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E13">A19</font></span></a>, <a class="txt" href="#E8:14_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="txt" href="#E9:14_3"><span class="lab"><font color="Green" title="E19">A25</font></span></a>, <a class="txt" href="#E12:14_3"><span class="lab"><font color="Green" title="E21">A27</font></span></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"> <font color="Maroon" title="c17">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c18">y</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="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">a</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E17:14_3"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">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"> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E19:14_3"/><span class="lab"><font color="Green" title="E22">A28</font></span>: 
<font color="Maroon" title="c19">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> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E20:14_3"/><span class="lab"><font color="Green" title="E23">A29</font></span>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">a12</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">x1</font>, <font color="Maroon" title="c21">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E22:14_3"/><span class="lab"><font color="Green" title="E24">A30</font></span>: 
( <font color="Maroon" title="c20">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c21">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> )
 <span class="kw">and </span><br/><a NAME="E23:14_3"/><span class="lab"><font color="Green" title="E25">A31</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c21">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:14_3"><span class="lab"><font color="Green" title="E22">A28</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E24:14_3"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c21">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:14_3"><span class="lab"><font color="Green" title="E24">A30</font></span></a>, <a class="ref" href="funct_3.html#D5" onmouseover="rs('funct_3/D5')" onmouseout="rh()">FUNCT_3:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:14_3"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14_3"><span class="lab"><font color="Green" title="E20">A26</font></span></a>, <a class="txt" href="#E20:14_3"><span class="lab"><font color="Green" title="E23">A29</font></span></a>, <a class="txt" href="#E23:14_3"><span class="lab"><font color="Green" title="E25">A31</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</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="E18:14"/>
<font color="Maroon" title="c11">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="14_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B2</font> or <font color="Maroon" title="c12">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="E1:14_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">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="c12">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="c13">w</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T12</font><span class="kw"> such that </span><br/><a NAME="E3:14_4"/><span class="lab"><font color="Green" title="E15">A32</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c13">w</font>)
 <span class="kw">and </span><br/><a NAME="E4:14_4"/><span class="lab"><font color="Green" title="E16">A33</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E13">A19</font></span></a></span>;<br/>
<a NAME="E5:14_4"/>
<font color="Maroon" title="c13">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="#E8:14"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E4:14_4"><span class="lab"><font color="Green" title="E16">A33</font></span></a></span>;<br/>
<a NAME="E6:14_4"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c13">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="ref" href="borsuk_1.html#T18" onmouseover="rs('borsuk_1/T18')" onmouseout="rh()">BORSUK_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:14_4"/>
<font color="Maroon" title="c12">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="#E3:14_4"><span class="lab"><font color="Green" title="E15">A32</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="c12">B2</font> = <font color="Maroon" title="c11">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="#E17:14"><span class="lab"><font color="Green" title="E14">A20</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="E20:14"/><span class="lab"><font color="Green" title="E15">A34</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>2</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="c7">T12</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="c8">B12</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>
<a NAME="E21:14"/>
 <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="c12">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="E22:14"/>
 <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> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E16:14"><span class="lab"><font color="Green" title="E13">A19</font></span></a>, <a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E15">A34</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>
