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

<b>set </b><font color="Maroon" title="c1">S</font> =  <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ;<br/>
<b>let </b><font color="Maroon" title="c2">X</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel18.html#D5" target="_self" title="WAYBEL18:def.5">WAYBEL18:def 5</a><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ;<br/>



<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<a NAME="E2:34_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<b>let </b><font color="Maroon" title="c4">Y</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>;<br/>

<b>assume </b><a NAME="E3:34_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c2">X</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">Y</font>
 ;<br/>

<a NAME="E4:34_1_1"/><b>then </b><i><font color="Green" title="E20">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35" title="BORSUK_1:th.35">BORSUK_1:35</a></i>;<br/>
<a NAME="E5:34_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T12" title="ZFMISC_1:th.12">ZFMISC_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">u</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>;<br/>
<a NAME="E7:34_1_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E8:34_1_1"/><b>then </b>
<font color="Maroon" title="c5">u</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 <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> 
 
<b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>;<br/>
<a NAME="E9:34_1_1"/><b>then </b>
<font color="Maroon" title="c5">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:34_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:34_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="waybel18.html#E2:34_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<a NAME="E11:34_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">u</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">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">Y</font><b> such that </b><br/><a NAME="E13:34_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c6">V</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="c4">Y</font>
 <b>and </b><br/><a NAME="E14:34_1_1"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:34_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">V</font> = <font color="Maroon" title="c6">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">Y</font> ;<br/>
<a NAME="E16:34_1_1"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c7">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E13:34_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">g</font> =  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c7">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y</font>;<br/>
<a NAME="E17:34_1_1"/><i><font color="Green" title="E24">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c7">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E18:34_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">g</font> =  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c7">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">Y</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c9">g</font>
;<br/>

<a NAME="E20:34_1_1"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>   st <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_1"/>
<font color="Maroon" title="c10">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:34_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">P</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 <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:34_1_1_1"/><b>then </b><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c10">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1_1_1"/>
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><b>by </b><i><a class="txt" href="waybel18.html#E3:34_1_1_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_1_1_1"/>
<font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T171" title="RELAT_1:th.171">RELAT_1:171</a></i>;<br/><a NAME="E3:34_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</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="c4">Y</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a></i>;<br/><b>hence </b><a NAME="E4:34_1_1_1_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_1_1_2"/><i><font color="Green" title="E26">A10</font></i>: 
<font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_1_2"/><i><font color="Green" title="E27">A11</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1_1_1_1_2_1"/><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font>
 ;<br/>

<a NAME="E2:34_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:34_1_1_1_1_2"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E3:34_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1_1_1_1_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:34_1_1_1_1_2_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:34_1_1_1_1_2"/>
<font color="Maroon" title="c7">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1_1_1_1_2_2"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 ;<br/>

<a NAME="E2:34_1_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E3:34_1_1_1_1_2_2"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="waybel18.html#E17:34_1_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="waybel18.html#E1:34_1_1_1_1_2_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1_1_1_1_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:34_1_1_1_1_2"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:34_1_1_1_1_2"/>
<font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E16:34_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="waybel18.html#E2:34_1_1_1_1_2"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:34_1_1_1_1_3"/>
<font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T48" title="FUNCT_2:th.48">FUNCT_2:48</a></i>;<br/><a NAME="E3:34_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</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="c4">Y</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/><b>hence </b><a NAME="E4:34_1_1_1_1_3"/>
<font color="Maroon" title="c9">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E21:34_1_1"/>
<font color="Maroon" title="c9">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:34_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>

<i><font color="Green" title="E25">A14</font></i>: <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>
<b>by </b><i><a class="txt" href="waybel18.html#E3:34_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="borsuk_1.html#T35" title="BORSUK_1:th.35">BORSUK_1:35</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E23:34_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_3"/><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<a NAME="E2:34_1_1_3"/><b>then </b><i><font color="Green" title="E27">A16</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1_3_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> or  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_3_1_1"/><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 ;<br/><div class="add"><a NAME="E2:34_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E29">A18</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/><a NAME="E3:34_1_1_3_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">u</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E14:34_1_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="waybel18.html#E1:34_1_1_3"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="waybel18.html#E1:34_1_1_3_1_1"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:34_1_1_3_1_1"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:34_1_1_3_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:34_1_1_3_1_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_3_1_2"/><i><font color="Green" title="E28">A19</font></i>: 
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 ;<br/><div class="add"><a NAME="E2:34_1_1_3_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:34_1_1_3"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:34_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> 
 ;<br/><a NAME="E4:34_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>;<br/><a NAME="E5:34_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E29">A20</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E6:34_1_1_3_1_2"/>
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="waybel18.html#E14:34_1_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="waybel18.html#E1:34_1_1_3_1_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:34_1_1_3_1_2"/><b>then </b>
(  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> or  not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E8:34_1_1_3_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E4:34_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="waybel18.html#E1:34_1_1_3"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="waybel18.html#E2:34_1_1_3"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="waybel18.html#E1:34_1_1_3_1_2"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="waybel18.html#E5:34_1_1_3_1_2"><i><font color="Green" title="E29">A20</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E24:34_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E22:34_1_1"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div>
