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

<b>let </b><font color="Maroon" title="c8">UL</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c8">UL</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c8">UL</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">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>let </b><font color="Maroon" title="c10">F</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c11">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c9">Y</font>;<br/>





<b>set </b><font color="Maroon" title="c12">L</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1;<br/>
<b>set </b><font color="Maroon" title="c13">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y</font>;<br/>
<a NAME="E3:38"/><i><font color="Green" title="E38">A3</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">UL</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:38"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a></i>;<br/>
<a NAME="E4:38"/><i><font color="Green" title="E39">A4</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>   ex <font color="Olive" title="b2">U</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Olive" title="b1">i</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> &amp; <font color="Olive" title="b2">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UL</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>consider </b><font color="Maroon" title="c15">U</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:38_1"/><i><font color="Green" title="E40">A5</font></i>: 
( <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c14">i</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">U</font> &amp; <font color="Maroon" title="c15">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UL</font> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:38"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">U</font> = <font color="Maroon" title="c15">U</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E2:38"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="topalg_5.html#E2:38_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">U</font>
;<br/>

<b>thus </b><a NAME="E4:38_1"/>
( <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c14">i</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">U</font> &amp; <font color="Maroon" title="c16">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UL</font> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:38_1"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:38"/><b>then </b>
 ex <font color="Olive" title="b1">U</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UL</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">UL1</font> = <font color="Maroon" title="c8">UL</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">V</font> being <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">UL1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y</font>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V</font> );<br/>
<a NAME="E7:38"/><i><font color="Green" title="E40">A6</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>   ex <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">UL1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<a NAME="E1:38_2"/>
 ex <font color="Olive" title="b1">U</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c15">i</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UL</font> )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:38"><i><font color="Green" title="E39">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:38_2"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">UL1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">i</font>,<font color="Olive" title="b1">z</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">I0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<font color="Maroon" title="c14">UL1</font><b> such that </b><br/><a NAME="E9:38"/><i><font color="Green" title="E41">A7</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c15">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="topalg_5.html#E7:38"><i><font color="Green" title="E40">A6</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">M</font> being <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="c9">Y</font> ex <font color="Olive" title="b2">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp; <font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> );<br/>
<a NAME="E10:38"/><i><font color="Green" title="E42">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>   ex <font color="Olive" title="b2">z</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>consider </b><font color="Maroon" title="c17">V</font> being  <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E2:38_3"/>
<font color="Maroon" title="c17">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">UL1</font>
 <b>and </b><br/><a NAME="E3:38_3"/><i><font color="Green" title="E43">A9</font></i>: 
<font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c16">i</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">V</font>
 <b>and </b><br/><a NAME="E4:38_3"/><i><font color="Green" title="E44">A10</font></i>: 
<font color="Maroon" title="c15">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:38"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E6:38_3"/><i><font color="Green" title="E45">A11</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c16">i</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">W</font>
 <b>and </b><br/><a NAME="E7:38_3"/><i><font color="Green" title="E46">A12</font></i>: 
<font color="Maroon" title="c18">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E8:38_3"/><i><font color="Green" title="E47">A13</font></i>: 
<font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c18">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:38_3"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">Q</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E10:38_3"/><i><font color="Green" title="E48">A14</font></i>: 
<font color="Maroon" title="c18">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c19">Q</font>
 <b>and </b><br/><a NAME="E11:38_3"/><i><font color="Green" title="E49">A15</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Q</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="c9">Y</font> ex <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; <font color="Olive" title="b2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E7:38_3"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="borsuk_1.html#T45" title="BORSUK_1:th.45">BORSUK_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:38_3"/><i><font color="Green" title="E50">A16</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c16">i</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Z</font>
 <b>and </b><br/><a NAME="E14:38_3"/><i><font color="Green" title="E51">A17</font></i>: 
<font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Q</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:38_3"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="topalg_5.html#E10:38_3"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">Y</font>, <font color="Maroon" title="c22">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E16:38_3"/><i><font color="Green" title="E52">A18</font></i>: 
<font color="Maroon" title="c20">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c21">A</font>,<font color="Maroon" title="c22">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>and </b><br/><a NAME="E17:38_3"/><i><font color="Green" title="E53">A19</font></i>: 
<font color="Maroon" title="c21">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E18:38_3"/><i><font color="Green" title="E54">A20</font></i>: 
<font color="Maroon" title="c22">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:38_3"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="topalg_5.html#E14:38_3"><i><font color="Green" title="E51">A17</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">A</font> = <font color="Maroon" title="c21">A</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="c9">Y</font> <b>by </b><i><a class="txt" href="topalg_5.html#E17:38_3"><i><font color="Green" title="E53">A19</font></i></a></i>;<br/>
<a NAME="E20:38_3"/><i><font color="Green" title="E55">A21</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:38_3"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="topalg_5.html#E16:38_3"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">B</font> = <font color="Maroon" title="c22">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E13:38_3"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="topalg_5.html#E16:38_3"><i><font color="Green" title="E52">A18</font></i></a>, <a class="txt" href="topalg_5.html#E18:38_3"><i><font color="Green" title="E54">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">i1</font> = <font color="Maroon" title="c16">i</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">B</font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E20:38_3"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E23:38_3"/><i><font color="Green" title="E56">A22</font></i>: 
 <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c24">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c25">i1</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:38_3"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="connsp_3.html#D7" title="CONNSP_3:def.7">CONNSP_3:def 7</a></i>;<br/>
<a NAME="E24:38_3"/>
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c25">i1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">B</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T43" title="CONNSP_1:th.43">CONNSP_1:43</a></i>;<br/>
<a NAME="E25:38_3"/><b>then </b>
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c25">i1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T21" title="CONNSP_2:th.21">CONNSP_2:21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">D</font> =  <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c24">B</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E20:38_3"><i><font color="Green" title="E55">A21</font></i></a>, <a class="txt" href="topalg_5.html#E23:38_3"><i><font color="Green" title="E56">A22</font></i></a>, <a class="ref" href="connsp_3.html#T35" title="CONNSP_3:th.35">CONNSP_3:35</a>, <a class="ref" href="tsep_1.html#T17" title="TSEP_1:th.17">TSEP_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">z</font> = <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c27">z</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c23">A</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c26">D</font>
;<br/>

<b>thus </b><a NAME="E28:38_3"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">A</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E13:38_3"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="topalg_5.html#E16:38_3"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>

<b>thus </b><a NAME="E29:38_3"/>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">D</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:38_3"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="connsp_3.html#T26" title="CONNSP_3:th.26">CONNSP_3:26</a></i>;<br/>

<a NAME="E30:38_3"/>
<font color="Maroon" title="c26">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">B</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:38_3"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="topalg_5.html#T3" target="_self" title="TOPALG_5:th.3">Th3</a></i>;<br/>
<a NAME="E31:38_3"/><b>then </b><i><font color="Green" title="E57">A23</font></i>: 
<font color="Maroon" title="c27">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c24">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E32:38_3"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c24">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">W</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10:38_3"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="topalg_5.html#E14:38_3"><i><font color="Green" title="E51">A17</font></i></a>, <a class="txt" href="topalg_5.html#E16:38_3"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E33:38_3"/><b>then </b>
<font color="Maroon" title="c27">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">W</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E31:38_3"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E34:38_3"/><b>then </b>
<font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c27">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c18">W</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E35:38_3"/>
<font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:38_3"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="topalg_5.html#E8:38_3"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>

<b>thus </b><a NAME="E36:38_3"/>
<font color="Maroon" title="c27">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c16">I1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> , <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E12:38"/><i><font color="Green" title="E43">A24</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c16">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="topalg_5.html#E10:38"><i><font color="Green" title="E42">A8</font></i></a>);<br/></i>
<a NAME="E13:38"/><i><font color="Green" title="E44">A25</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">I1</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">C1</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">I1</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E15:38"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">C1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">a</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="E1:38_4"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">a1</font>, <font color="Maroon" title="c20">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_4"/><i><font color="Green" title="E45">A26</font></i>: 
<font color="Maroon" title="c19">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E4:38_4"/><i><font color="Green" title="E46">A27</font></i>: 
<font color="Maroon" title="c20">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <b>and </b><br/><a NAME="E5:38_4"/><i><font color="Green" title="E47">A28</font></i>: 
<font color="Maroon" title="c18">a</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="c19">a1</font>,<font color="Maroon" title="c20">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">a2</font> = <font color="Maroon" title="c20">a2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E4:38_4"><i><font color="Green" title="E46">A27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">M</font> being  <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="c9">Y</font>, <font color="Maroon" title="c23">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E8:38_4"/><i><font color="Green" title="E48">A29</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">M</font> &amp; <font color="Maroon" title="c21">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">O</font> )
 <b>and </b><br/><a NAME="E9:38_4"/>
<font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c22">M</font>,<font color="Maroon" title="c23">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">a2</font>
 <b>and </b><br/><a NAME="E10:38_4"/><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c22">M</font>,<font color="Maroon" title="c23">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:38"><i><font color="Green" title="E43">A24</font></i></a></i>;<br/>
<a NAME="E11:38_4"/><i><font color="Green" title="E50">A31</font></i>: 
<font color="Maroon" title="c19">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:38_4"><i><font color="Green" title="E45">A26</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:38_4"/><i><font color="Green" title="E51">A32</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c21">a2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c22">M</font>,<font color="Maroon" title="c23">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8:38_4"><i><font color="Green" title="E48">A29</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:38_4"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c22">M</font>,<font color="Maroon" title="c23">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">C1</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:38"><i><font color="Green" title="E44">A25</font></i></a>, <a class="txt" href="topalg_5.html#E10:38_4"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E14:38_4"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">C1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:38_4"><i><font color="Green" title="E47">A28</font></i></a>, <a class="txt" href="topalg_5.html#E11:38_4"><i><font color="Green" title="E50">A31</font></i></a>, <a class="txt" href="topalg_5.html#E12:38_4"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:38"/><b>then </b><i><font color="Green" title="E45">A33</font></i>: 
<font color="Maroon" title="c17">C1</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>;<br/>
<a NAME="E17:38"/><i><font color="Green" title="E46">A34</font></i>: 
<font color="Maroon" title="c17">C1</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:38_5"/>
<font color="Maroon" title="c18">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">C1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_5"/><i><font color="Green" title="E47">A35</font></i>: 
<font color="Maroon" title="c19">i</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="c16">I1</font>
 <b>and </b><br/><a NAME="E4:38_5"/><i><font color="Green" title="E48">A36</font></i>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">P</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:38_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>]
 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:38"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="topalg_5.html#E3:38_5"><i><font color="Green" title="E47">A35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:38_5"/>
<font color="Maroon" title="c18">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:38_5"><i><font color="Green" title="E48">A36</font></i></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:38"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T27" title="BORSUK_3:th.27">BORSUK_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E20:38"/><i><font color="Green" title="E47">A37</font></i>: 
<font color="Maroon" title="c18">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">C1</font>
 <b>and </b><br/><a NAME="E21:38"/><i><font color="Green" title="E48">A38</font></i>: 
<font color="Maroon" title="c18">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>and </b><br/><a NAME="E22:38"/><i><font color="Green" title="E49">A39</font></i>: 
<font color="Maroon" title="c18">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:38"><i><font color="Green" title="E45">A33</font></i></a>, <a class="txt" href="topalg_5.html#E17:38"><i><font color="Green" title="E46">A34</font></i></a>, <a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a></i>;<br/>
<a NAME="E23:38"/><i><font color="Green" title="E50">A40</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c18">G</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E21:38"><i><font color="Green" title="E48">A38</font></i></a>, <a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>;<br/>
<a NAME="E24:38"/><i><font color="Green" title="E51">A41</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">ICOV</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  :  ex <font color="Olive" title="b2">M</font> being <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="c9">Y</font> st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> </span>}</span> ;<br/>
<a NAME="E25:38"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  :  ex <font color="Olive" title="b2">M</font> being <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="c9">Y</font> st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">a</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="E1:38_6"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  :  ex <font color="Olive" title="b2">M</font> being <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="c9">Y</font> st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> </span>}</span> 
 ;<br/>

<a NAME="E2:38_6"/><b>then </b>
 ex <font color="Olive" title="b1">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">O</font> &amp;  ex <font color="Olive" title="b2">M</font> being <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="c9">Y</font> st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<b>hence </b><a NAME="E3:38_6"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c20">ICOV</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  :  ex <font color="Olive" title="b2">M</font> being <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="c9">Y</font> st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E27:38"/><i><font color="Green" title="E52">A42</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c20">ICOV</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">a</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="E1:38_7"/>
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c22">a</font> = <font color="Maroon" title="c21">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E3:38_7"/>
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c22">a</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E24:38"><i><font color="Green" title="E51">A41</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:38_7"/><i><font color="Green" title="E53">A43</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c22">a</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Z</font>
 <b>and </b><br/><a NAME="E6:38_7"/><i><font color="Green" title="E54">A44</font></i>: 
<font color="Maroon" title="c23">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E23:38"><i><font color="Green" title="E50">A40</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:38_7"/><i><font color="Green" title="E55">A45</font></i>: 
<font color="Maroon" title="c24">i</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="c16">I1</font>
 <b>and </b><br/><a NAME="E9:38_7"/><i><font color="Green" title="E56">A46</font></i>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Z</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:38"><i><font color="Green" title="E47">A37</font></i></a>, <a class="txt" href="topalg_5.html#E6:38_7"><i><font color="Green" title="E54">A44</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">M</font> being  <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="c9">Y</font>, <font color="Maroon" title="c26">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E11:38_7"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">M</font>
 <b>and </b><br/><a NAME="E12:38_7"/>
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">O</font>
 <b>and </b><br/><a NAME="E13:38_7"/>
<font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c25">M</font>,<font color="Maroon" title="c26">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
 <b>and </b><br/><a NAME="E14:38_7"/><i><font color="Green" title="E57">A47</font></i>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c25">M</font>,<font color="Maroon" title="c26">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:38"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="topalg_5.html#E8:38_7"><i><font color="Green" title="E55">A45</font></i></a></i>;<br/>
<a NAME="E15:38_7"/><i><font color="Green" title="E58">A48</font></i>: 
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">O</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:38_7"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="topalg_5.html#E9:38_7"><i><font color="Green" title="E56">A46</font></i></a>, <a class="txt" href="topalg_5.html#E14:38_7"><i><font color="Green" title="E57">A47</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E16:38_7"/>
<font color="Maroon" title="c26">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ICOV</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E6:38_7"><i><font color="Green" title="E54">A44</font></i></a>, <a class="txt" href="topalg_5.html#E9:38_7"><i><font color="Green" title="E56">A46</font></i></a>, <a class="txt" href="topalg_5.html#E14:38_7"><i><font color="Green" title="E57">A47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:38_7"/>
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c20">ICOV</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:38_7"><i><font color="Green" title="E58">A48</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:38"/><i><font color="Green" title="E53">A49</font></i>: 
<font color="Maroon" title="c20">ICOV</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_8"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_8"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c20">ICOV</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E27:38"><i><font color="Green" title="E52">A42</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:38_8"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c20">ICOV</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E29:38"/><i><font color="Green" title="E54">A50</font></i>: 
<font color="Maroon" title="c20">ICOV</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">a</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:38_9"/>
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ICOV</font>
 ;<br/>

<a NAME="E2:38_9"/><b>then </b>
 ex <font color="Olive" title="b1">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Maroon" title="c21">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">O</font> &amp;  ex <font color="Olive" title="b2">M</font> being <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="c9">Y</font> st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<b>hence </b><a NAME="E3:38_9"/>
<font color="Maroon" title="c21">a</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c21">NCOV</font> being    <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Maroon" title="c20">ICOV</font>;<br/>
<b>consider </b><font color="Maroon" title="c22">T</font> being    <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Maroon" title="c21">NCOV</font>;<br/>
<a NAME="E32:38"/><i><font color="Green" title="E55">A51</font></i>: 
<font color="Maroon" title="c20">ICOV</font> is <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rcomp_3.html#D1" title="RCOMP_3:def.1">RCOMP_3:def 1</a><br/>

<b>assume </b><a NAME="E1:38_10"/>
<font color="Maroon" title="c23">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ICOV</font>
 ;<br/>

<a NAME="E2:38_10"/><b>then </b>
 ex <font color="Olive" title="b1">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Maroon" title="c23">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">O</font> &amp;  ex <font color="Olive" title="b2">M</font> being <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="c9">Y</font> st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<b>hence </b><a NAME="E3:38_10"/>
<font color="Maroon" title="c23">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c23">T</font> = <font color="Maroon" title="c22">T</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E28:38"><i><font color="Green" title="E53">A49</font></i></a>, <a class="txt" href="topalg_5.html#E29:38"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="topalg_5.html#E40"><i><font color="Green" title="E34">Lm15</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c23">T</font>
;<br/>

<b>thus </b><a NAME="E34:38"/>
( <font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="topalg_5.html#E28:38"><i><font color="Green" title="E53">A49</font></i></a>, <a class="txt" href="topalg_5.html#E29:38"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="topalg_5.html#E32:38"><i><font color="Green" title="E55">A51</font></i></a>, <a class="ref" href="rcomp_3.html#D3" title="RCOMP_3:def.3">RCOMP_3:def 3</a></i>;<br/>

<b>thus </b><a NAME="E35:38"/>
<font color="Maroon" title="c23">T</font> is <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E28:38"><i><font color="Green" title="E53">A49</font></i></a>, <a class="txt" href="topalg_5.html#E29:38"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="topalg_5.html#E32:38"><i><font color="Green" title="E55">A51</font></i></a>, <a class="ref" href="rcomp_3.html#T79" title="RCOMP_3:th.79">RCOMP_3:79</a></i>;<br/>

<b>set </b><font color="Maroon" title="c24">NN</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">M</font> where <font color="Olive" title="b1">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">Y</font> : ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> ) </span>}</span> ;<br/>
<a NAME="E36:38"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">M</font> where <font color="Olive" title="b1">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">Y</font> : ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">a</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="E1:38_11"/>
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">M</font> where <font color="Olive" title="b1">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">Y</font> : ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:38_11"/><b>then </b>
 ex <font color="Olive" title="b1">M</font> being <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="c9">Y</font> st <br/>( <font color="Maroon" title="c25">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<b>hence </b><a NAME="E3:38_11"/>
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c25">NN</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">M</font> where <font color="Olive" title="b1">M</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">Y</font> : ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c9">Y</font> ;<br/>
<a NAME="E38:38"/>
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E24:38"><i><font color="Green" title="E51">A41</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E40:38"/>
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">Z</font>
 <b>and </b><br/><a NAME="E41:38"/><i><font color="Green" title="E56">A52</font></i>: 
<font color="Maroon" title="c26">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E23:38"><i><font color="Green" title="E50">A40</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E43:38"/><i><font color="Green" title="E57">A53</font></i>: 
<font color="Maroon" title="c27">i</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="c16">I1</font>
 <b>and </b><br/><a NAME="E44:38"/><i><font color="Green" title="E58">A54</font></i>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Z</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:38"><i><font color="Green" title="E47">A37</font></i></a>, <a class="txt" href="topalg_5.html#E41:38"><i><font color="Green" title="E56">A52</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">M</font> being  <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="c9">Y</font>, <font color="Maroon" title="c29">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E46:38"/><i><font color="Green" title="E59">A55</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">M</font>
 <b>and </b><br/><a NAME="E47:38"/>
( <font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">O</font> &amp; <font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c28">M</font>,<font color="Maroon" title="c29">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> )
 <b>and </b><br/><a NAME="E48:38"/><i><font color="Green" title="E60">A56</font></i>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c28">M</font>,<font color="Maroon" title="c29">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:38"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="topalg_5.html#E43:38"><i><font color="Green" title="E57">A53</font></i></a></i>;<br/>
<a NAME="E49:38"/><i><font color="Green" title="E61">A57</font></i>: 
<font color="Maroon" title="c28">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">NN</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E41:38"><i><font color="Green" title="E56">A52</font></i></a>, <a class="txt" href="topalg_5.html#E44:38"><i><font color="Green" title="E58">A54</font></i></a>, <a class="txt" href="topalg_5.html#E46:38"><i><font color="Green" title="E59">A55</font></i></a>, <a class="txt" href="topalg_5.html#E48:38"><i><font color="Green" title="E60">A56</font></i></a></i>;<br/>
<a NAME="E50:38"/><i><font color="Green" title="E62">A58</font></i>: 
<font color="Maroon" title="c25">NN</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">a</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:38_12"/>
<font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">NN</font>
 ;<br/>

<a NAME="E2:38_12"/><b>then </b>
 ex <font color="Olive" title="b1">M</font> being <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="c9">Y</font> st <br/>( <font color="Maroon" title="c30">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<b>hence </b><a NAME="E3:38_12"/>
<font color="Maroon" title="c30">a</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c30">p</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E52:38"/><i><font color="Green" title="E63">A59</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G</font>
 <b>and </b><br/><a NAME="E53:38"/><i><font color="Green" title="E64">A60</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E22:38"><i><font color="Green" title="E49">A39</font></i></a>, <a class="ref" href="finset_1.html#D1" title="FINSET_1:def.1">FINSET_1:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">M</font> being <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="c9">Y</font> ex <font color="Olive" title="b2">O</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c30">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> );<br/>
<a NAME="E54:38"/><i><font color="Green" title="E65">A61</font></i>: 
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="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">p</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">NN</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c31">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_13"/>
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">p</font>
 ;<br/>

<a NAME="E2:38_13"/><b>then </b><i><font color="Green" title="E66">A62</font></i>: 
<font color="Maroon" title="c30">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">p</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c32">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:38_13"/><i><font color="Green" title="E67">A63</font></i>: 
<font color="Maroon" title="c32">i</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="c16">I1</font>
 <b>and </b><br/><a NAME="E5:38_13"/><i><font color="Green" title="E68">A64</font></i>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:38"><i><font color="Green" title="E47">A37</font></i></a>, <a class="txt" href="topalg_5.html#E52:38"><i><font color="Green" title="E63">A59</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c33">M</font> being  <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="c9">Y</font>, <font color="Maroon" title="c34">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E7:38_13"/><i><font color="Green" title="E69">A65</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">M</font> &amp; <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">O</font> &amp; <font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c33">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> &amp; <font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c33">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:38"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="topalg_5.html#E4:38_13"><i><font color="Green" title="E67">A63</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c33">M</font>
;<br/>

<b>thus </b><a NAME="E8:38_13"/>
( <font color="Maroon" title="c33">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">NN</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c31">x</font>,<font color="Maroon" title="c33">M</font>] )
 <b>by </b><i><a class="txt" href="topalg_5.html#E52:38"><i><font color="Green" title="E63">A59</font></i></a>, <a class="txt" href="topalg_5.html#E2:38_13"><i><font color="Green" title="E66">A62</font></i></a>, <a class="txt" href="topalg_5.html#E5:38_13"><i><font color="Green" title="E68">A64</font></i></a>, <a class="txt" href="topalg_5.html#E7:38_13"><i><font color="Green" title="E69">A65</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c31">p1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">p</font>,<font color="Maroon" title="c25">NN</font><b> such that </b><br/><a NAME="E56:38"/><i><font color="Green" title="E66">A66</font></i>: 
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="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">p</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c31">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="topalg_5.html#E54:38"><i><font color="Green" title="E65">A61</font></i></a>);<br/></i>
<a NAME="E57:38"/><i><font color="Green" title="E67">A67</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c31">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">p</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E49:38"><i><font color="Green" title="E61">A57</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E58:38"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c31">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">NN</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_14"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_14"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c31">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">NN</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c32">a</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="E2:38_14"/>
<font color="Maroon" title="c32">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">NN</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c33">M</font> being  <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="c9">Y</font><b> such that </b><br/><a NAME="E4:38_14"/><i><font color="Green" title="E68">A68</font></i>: 
<font color="Maroon" title="c32">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">M</font>
 <b>and </b><br/><a NAME="E5:38_14"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">M</font>
 <b>and </b><br/><a NAME="E6:38_14"/><i><font color="Green" title="E69">A69</font></i>: 
 ex <font color="Olive" title="b1">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c33">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c34">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E8:38_14"/><i><font color="Green" title="E70">A70</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c33">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:38_14"><i><font color="Green" title="E69">A69</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c35">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:38_14"/><i><font color="Green" title="E71">A71</font></i>: 
<font color="Maroon" title="c35">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c30">p</font>
 <b>and </b><br/><a NAME="E11:38_14"/><i><font color="Green" title="E72">A72</font></i>: 
<font color="Maroon" title="c30">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c33">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E52:38"><i><font color="Green" title="E63">A59</font></i></a>, <a class="txt" href="topalg_5.html#E8:38_14"><i><font color="Green" title="E70">A70</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c36">M1</font> being  <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="c9">Y</font>, <font color="Maroon" title="c37">O1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E13:38_14"/><i><font color="Green" title="E73">A73</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">M1</font>
 <b>and </b><br/><a NAME="E14:38_14"/><i><font color="Green" title="E74">A74</font></i>: 
<font color="Maroon" title="c30">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c36">M1</font>,<font color="Maroon" title="c37">O1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>and </b><br/><a NAME="E15:38_14"/><i><font color="Green" title="E75">A75</font></i>: 
<font color="Maroon" title="c31">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">M1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E56:38"><i><font color="Green" title="E66">A66</font></i></a>, <a class="txt" href="topalg_5.html#E10:38_14"><i><font color="Green" title="E71">A71</font></i></a></i>;<br/>
<a NAME="E16:38_14"/>
<font color="Maroon" title="c36">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">M</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:38_14"><i><font color="Green" title="E72">A72</font></i></a>, <a class="txt" href="topalg_5.html#E13:38_14"><i><font color="Green" title="E73">A73</font></i></a>, <a class="txt" href="topalg_5.html#E14:38_14"><i><font color="Green" title="E74">A74</font></i></a>, <a class="ref" href="zfmisc_1.html#T134" title="ZFMISC_1:th.134">ZFMISC_1:134</a></i>;<br/>
<b>hence </b><a NAME="E17:38_14"/>
<font color="Maroon" title="c32">a</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="c31">p1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E57:38"><i><font color="Green" title="E67">A67</font></i></a>, <a class="txt" href="topalg_5.html#E4:38_14"><i><font color="Green" title="E68">A68</font></i></a>, <a class="txt" href="topalg_5.html#E10:38_14"><i><font color="Green" title="E71">A71</font></i></a>, <a class="txt" href="topalg_5.html#E15:38_14"><i><font color="Green" title="E75">A75</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E59:38"/><b>then </b>
<font color="Maroon" title="c25">NN</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E53:38"><i><font color="Green" title="E64">A60</font></i></a>, <a class="txt" href="topalg_5.html#E57:38"><i><font color="Green" title="E67">A67</font></i></a>, <a class="ref" href="finset_1.html#D1" title="FINSET_1:def.1">FINSET_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c32">N</font> =  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c25">NN</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="c9">Y</font> <b>by </b><i><a class="txt" href="topalg_5.html#E50:38"><i><font color="Green" title="E62">A58</font></i></a>, <a class="ref" href="tops_2.html#T27" title="TOPS_2:th.27">TOPS_2:27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c32">N</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c33">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:38_15"/>
<font color="Maroon" title="c33">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">NN</font>
 ;<br/><a NAME="E2:38_15"/><b>then </b>
 ex <font color="Olive" title="b1">M</font> being <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="c9">Y</font> st <br/>( <font color="Maroon" title="c33">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 ;<br/><b>hence </b><a NAME="E3:38_15"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Z</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E62:38"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">N</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E49:38"><i><font color="Green" title="E61">A57</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>

<b>let </b><font color="Maroon" title="c33">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E63:38"/><i><font color="Green" title="E68">A76</font></i>: 
<font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">T</font>
 <b>and </b><br/><a NAME="E64:38"/><i><font color="Green" title="E69">A77</font></i>: 
<font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">T</font>
 ;<br/>

<a NAME="E65:38"/><i><font color="Green" title="E70">A78</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">i</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E63:38"><i><font color="Green" title="E68">A76</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E66:38"/><i><font color="Green" title="E71">A79</font></i>: 
<font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">T</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E64:38"><i><font color="Green" title="E69">A77</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E67:38"/><b>then </b><i><font color="Green" title="E72">A80</font></i>: 
<font color="Maroon" title="c33">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">T</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E68:38"/><i><font color="Green" title="E73">A81</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">NCOV</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">ICOV</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E28:38"><i><font color="Green" title="E53">A49</font></i></a>, <a class="txt" href="topalg_5.html#E29:38"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="topalg_5.html#E32:38"><i><font color="Green" title="E55">A51</font></i></a>, <a class="ref" href="rcomp_3.html#D2" title="RCOMP_3:def.2">RCOMP_3:def 2</a></i>;<br/>
<a NAME="E69:38"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">NCOV</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="topalg_5.html#E28:38"><i><font color="Green" title="E53">A49</font></i></a>, <a class="txt" href="topalg_5.html#E29:38"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="topalg_5.html#E32:38"><i><font color="Green" title="E55">A51</font></i></a>, <a class="ref" href="rcomp_3.html#D3" title="RCOMP_3:def.3">RCOMP_3:def 3</a></i>;<br/>
<a NAME="E70:38"/><b>then </b>
<font color="Maroon" title="c33">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">NCOV</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E66:38"><i><font color="Green" title="E71">A79</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E71:38"/><b>then </b>
<font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">NCOV</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E65:38"><i><font color="Green" title="E70">A78</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E72:38"/><b>then </b>
<font color="Maroon" title="c21">NCOV</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</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="c21">NCOV</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E73:38"/><b>then </b>
<font color="Maroon" title="c21">NCOV</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ICOV</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E68:38"><i><font color="Green" title="E73">A81</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c34">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E75:38"/><i><font color="Green" title="E74">A82</font></i>: 
<font color="Maroon" title="c21">NCOV</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">O</font>
 <b>and </b><br/><a NAME="E76:38"/><i><font color="Green" title="E75">A83</font></i>: 
 ex <font color="Olive" title="b1">M</font> being <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="c9">Y</font> st <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c35">M</font> being  <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="c9">Y</font><b> such that </b><br/><a NAME="E78:38"/><i><font color="Green" title="E76">A84</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c35">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E76:38"><i><font color="Green" title="E75">A83</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c36">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E80:38"/><i><font color="Green" title="E77">A85</font></i>: 
<font color="Maroon" title="c36">j</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="c16">I1</font>
 <b>and </b><br/><a NAME="E81:38"/><i><font color="Green" title="E78">A86</font></i>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c35">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:38"><i><font color="Green" title="E47">A37</font></i></a>, <a class="txt" href="topalg_5.html#E78:38"><i><font color="Green" title="E76">A84</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c37">M1</font> being  <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="c9">Y</font>, <font color="Maroon" title="c38">O1</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E83:38"/><i><font color="Green" title="E79">A87</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">M1</font>
 <b>and </b><br/><a NAME="E84:38"/><i><font color="Green" title="E80">A88</font></i>: 
<font color="Maroon" title="c36">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">O1</font>
 <b>and </b><br/><a NAME="E85:38"/><i><font color="Green" title="E81">A89</font></i>: 
<font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c37">M1</font>,<font color="Maroon" title="c38">O1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font>
 <b>and </b><br/><a NAME="E86:38"/><i><font color="Green" title="E82">A90</font></i>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c37">M1</font>,<font color="Maroon" title="c38">O1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:38"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="topalg_5.html#E80:38"><i><font color="Green" title="E77">A85</font></i></a></i>;<br/>
<a NAME="E87:38"/><i><font color="Green" title="E83">A91</font></i>: 
( <font color="Maroon" title="c35">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">M1</font> &amp; <font color="Maroon" title="c34">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">O1</font> )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E81:38"><i><font color="Green" title="E78">A86</font></i></a>, <a class="txt" href="topalg_5.html#E83:38"><i><font color="Green" title="E79">A87</font></i></a>, <a class="txt" href="topalg_5.html#E84:38"><i><font color="Green" title="E80">A88</font></i></a>, <a class="txt" href="topalg_5.html#E86:38"><i><font color="Green" title="E82">A90</font></i></a>, <a class="ref" href="zfmisc_1.html#T134" title="ZFMISC_1:th.134">ZFMISC_1:134</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c39">V</font> being  <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E89:38"/><i><font color="Green" title="E84">A92</font></i>: 
<font color="Maroon" title="c39">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">UL1</font>
 <b>and </b><br/><a NAME="E90:38"/><i><font color="Green" title="E85">A93</font></i>: 
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c36">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">V</font>
 <b>and </b><br/><a NAME="E91:38"/><i><font color="Green" title="E86">A94</font></i>: 
<font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:38"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="topalg_5.html#E80:38"><i><font color="Green" title="E77">A85</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c40">V</font> = <font color="Maroon" title="c39">V</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E90:38"><i><font color="Green" title="E85">A93</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c40">V</font>
;<br/>

<b>thus </b><a NAME="E93:38"/>
<font color="Maroon" title="c40">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">UL</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E89:38"><i><font color="Green" title="E84">A92</font></i></a></i>;<br/>

<a NAME="E94:38"/><i><font color="Green" title="E87">A95</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">O</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E28:38"><i><font color="Green" title="E53">A49</font></i></a>, <a class="txt" href="topalg_5.html#E29:38"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="topalg_5.html#E32:38"><i><font color="Green" title="E55">A51</font></i></a>, <a class="txt" href="topalg_5.html#E65:38"><i><font color="Green" title="E70">A78</font></i></a>, <a class="txt" href="topalg_5.html#E67:38"><i><font color="Green" title="E72">A80</font></i></a>, <a class="txt" href="topalg_5.html#E75:38"><i><font color="Green" title="E74">A82</font></i></a>, <a class="ref" href="rcomp_3.html#T80" title="RCOMP_3:th.80">RCOMP_3:80</a></i>;<br/>
<a NAME="E95:38"/>
<font color="Maroon" title="c35">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">NN</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E78:38"><i><font color="Green" title="E76">A84</font></i></a>, <a class="txt" href="topalg_5.html#E83:38"><i><font color="Green" title="E79">A87</font></i></a>, <a class="txt" href="topalg_5.html#E87:38"><i><font color="Green" title="E83">A91</font></i></a></i>;<br/>
<a NAME="E96:38"/><b>then </b>
<font color="Maroon" title="c32">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c35">M</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<a NAME="E97:38"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c32">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c37">M1</font>,<font color="Maroon" title="c38">O1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E81:38"><i><font color="Green" title="E78">A86</font></i></a>, <a class="txt" href="topalg_5.html#E86:38"><i><font color="Green" title="E82">A90</font></i></a>, <a class="txt" href="topalg_5.html#E94:38"><i><font color="Green" title="E87">A95</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E98:38"/><b>then </b>
<font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c32">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c37">M1</font>,<font color="Maroon" title="c38">O1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E99:38"/>
<font color="Maroon" title="c10">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c32">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">T</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c40">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E85:38"><i><font color="Green" title="E81">A89</font></i></a>, <a class="txt" href="topalg_5.html#E91:38"><i><font color="Green" title="E86">A94</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div>
