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

<b>let </b><font color="Maroon" title="c8">x0</font>, <font color="Maroon" title="c9">y0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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>let </b><font color="Maroon" title="c10">xt</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/><b>let </b><font color="Maroon" title="c11">f</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font>;<br/>





<b>assume </b><a NAME="E1:40"/>
<font color="Maroon" title="c10">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:40"/><b>then </b><i><font color="Green" title="E38">A1</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">xt</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="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">Y</font> =  <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E3:40"/><i><font color="Green" title="E39">A2</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <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> ) <b>-&gt; </b>   <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> = <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $2;<br/>
<b>consider </b><font color="Maroon" title="c13">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="E5:40"/><i><font color="Green" title="E40">A3</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span><br/> for <font color="Olive" title="b2">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" title="c13">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">i</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E6:40"/><i><font color="Green" title="E41">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,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>then reconsider </b><font color="Maroon" title="c14">F</font> = <font color="Maroon" title="c13">F</font> as   <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"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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>reconsider </b><font color="Maroon" title="c15">j</font> = 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:40"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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><br/> for <font color="Olive" title="b2">V</font> being  <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 <font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> ex <font color="Olive" title="b3">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"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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>;<br/><b>let </b><font color="Maroon" title="c17">V</font> be   <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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40_1"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">V</font>
 <b>and </b><br/><a NAME="E2:40_1"/><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon" title="c17">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c18">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <font color="Maroon" title="c19">p2</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> <b> such that </b><br/><a NAME="E4:40_1"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c19">p2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<a NAME="E5:40_1"/>
<font color="Maroon" title="c14">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">p1</font>,<font color="Maroon" title="c19">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">p2</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">S</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="E7:40_1"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c19">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">S</font>
 <b>and </b><br/><a NAME="E8:40_1"/><i><font color="Green" title="E46">A9</font></i>: 
<font color="Maroon" title="c20">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E9:40_1"/><i><font color="Green" title="E47">A10</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">S</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#E1:40_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>set </b><font color="Maroon" title="c21">W</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c20">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E10:40_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c20">S</font></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="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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,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="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">W</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c20">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <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"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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/>
<b>take </b>
<font color="Maroon" title="c22">W</font>
;<br/>

<b>thus </b><a NAME="E12:40_1"/>
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_1"><i><font color="Green" title="E45">A8</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="E13:40_1"/>
<font color="Maroon" title="c22">W</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#E3:40"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_1"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon" title="c23">y</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="E14:40_1"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c22">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:40_1"/><i><font color="Green" title="E48">A11</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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>and </b><br/><a NAME="E17:40_1"/><i><font color="Green" title="E49">A12</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W</font>
 <b>and </b><br/><a NAME="E18:40_1"/><i><font color="Green" title="E50">A13</font></i>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <font color="Maroon" title="c26">x2</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> <b> such that </b><br/><a NAME="E20:40_1"/><i><font color="Green" title="E51">A14</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c25">x1</font>,<font color="Maroon" title="c26">x2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:40_1"><i><font color="Green" title="E48">A11</font></i></a>, <a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<i><font color="Green" title="E52">A15</font></i>: <font color="Maroon" title="c23">y</font> = 
<font color="Maroon" title="c14">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c25">x1</font>,<font color="Maroon" title="c26">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E20:40_1"><i><font color="Green" title="E51">A14</font></i></a>, <a class="txt" href="topalg_5.html#E18:40_1"><i><font color="Green" title="E50">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40"><i><font color="Green" title="E40">A3</font></i></a></i>
;<br/>
<a NAME="E22:40_1"/>
<font color="Maroon" title="c26">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">S</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E17:40_1"><i><font color="Green" title="E49">A12</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_1"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E23:40_1"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E24:40_1"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:40_1"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_1"><i><font color="Green" title="E52">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:40"/><b>then </b><i><font color="Green" title="E42">A16</font></i>: 
<font color="Maroon" title="c14">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E11:40"/><i><font color="Green" title="E43">A17</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">F</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 <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:40"/><b>then </b><i><font color="Green" title="E44">A18</font></i>: 
<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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">F</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E13:40"/><b>then </b><i><font color="Green" title="E45">A19</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><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 <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E14:40"/><i><font color="Green" title="E46">A20</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></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>set </b><font color="Maroon" title="c16">Ft</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font>;<br/>
<a NAME="E15:40"/><i><font color="Green" title="E47">A21</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="txt" href="topalg_5.html#E14:40"><i><font color="Green" title="E46">A20</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="E16:40"/><i><font color="Green" title="E48">A22</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span><br/> for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">i</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">xt</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E14:40"><i><font color="Green" title="E46">A20</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a></i>;<br/>
<a NAME="E17:40"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><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 <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_2"/><i><font color="Green" title="E49">A23</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c18">x1</font>, <font color="Maroon" title="c19">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_2"/><i><font color="Green" title="E50">A24</font></i>: 
<font color="Maroon" title="c18">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2"/><i><font color="Green" title="E51">A25</font></i>: 
<font color="Maroon" title="c19">x2</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">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E5:40_2"/><i><font color="Green" title="E52">A26</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2"><i><font color="Green" title="E49">A23</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="txt" href="topalg_5.html#E14:40"><i><font color="Green" title="E46">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:40_2"/><i><font color="Green" title="E53">A27</font></i>: 
<font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green" title="E51">A25</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><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 <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> = 
<font color="Maroon" title="c14">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2"><i><font color="Green" title="E49">A23</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="txt" href="topalg_5.html#E14:40"><i><font color="Green" title="E46">A20</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green" title="E53">A27</font></i></a>, <a class="txt" href="topalg_5.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">x0</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green" title="E53">A27</font></i></a>, <a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">xt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E16:40"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green" title="E51">A25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2"><i><font color="Green" title="E49">A23</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E18:40"/><b>then </b>
<font color="Maroon" title="c14">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="topalg_5.html#E15:40"><i><font color="Green" title="E47">A21</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">G</font> being   <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"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> <b> such that </b><br/><a NAME="E20:40"/><i><font color="Green" title="E49">A28</font></i>: 
<font color="Maroon" title="c17">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E21:40"/><i><font color="Green" title="E50">A29</font></i>: 
<font color="Maroon" title="c14">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font>
 <b>and </b><br/><a NAME="E22:40"/><i><font color="Green" title="E51">A30</font></i>: 
<font color="Maroon" title="c17">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font>
 <b>and </b><br/><a NAME="E23:40"/><i><font color="Green" title="E52">A31</font></i>: 
for <font color="Olive" title="b1">H</font> being  <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"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c14">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font> holds <br/><font color="Maroon" title="c17">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:40"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="topalg_5.html#T22" target="_self" title="TOPALG_5:th.22">Th22</a></i>;<br/>
<b>take </b><font color="Maroon" title="c18">ft</font> =  <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c17">G</font>;<br/>

<a NAME="E24:40"/><i><font color="Green" title="E53">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="c15">j</font>,<font color="Maroon" title="c4">j0</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="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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green" title="E4">Lm4</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><font color="Maroon" title="c18">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 = 
<font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c4">j0</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c4">j0</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E22:40"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="topalg_5.html#E24:40"><i><font color="Green" title="E53">A32</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c10">xt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E16:40"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>
;<br/>

<a NAME="E26:40"/>
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>  holds  <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ft</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">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/>

<a NAME="E1:40_4"/><i><font color="Green" title="E54">A33</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,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>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ft</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c19">i</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c19">i</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_4"><i><font color="Green" title="E54">A33</font></i></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E21:40"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green" title="E40">A3</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E27:40"/>
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">ft</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>

<a NAME="E28:40"/>
for <font color="Olive" title="b1">H</font> being <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"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  holds   <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c15">j</font>,<font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
;<br/>
<b>hence </b><a NAME="E29:40"/>
<font color="Maroon" title="c18">ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:40"><i><font color="Green" title="E49">A28</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c19">f1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E30:40"/><i><font color="Green" title="E54">A34</font></i>: 
<font color="Maroon" title="c19">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E31:40"/><i><font color="Green" title="E55">A35</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">f1</font>
 <b>and </b><br/><a NAME="E32:40"/><i><font color="Green" title="E56">A36</font></i>: 
<font color="Maroon" title="c19">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">xt</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <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> ) <b>-&gt; </b>   <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="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  = <font color="Maroon" title="c19">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $2;<br/>
<b>consider </b><font color="Maroon" title="c20">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> <b> such that </b><br/><a NAME="E34:40"/><i><font color="Green" title="E57">A37</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span><br/> for <font color="Olive" title="b2">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" title="c20">H</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">i</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c21">H</font> = <font color="Maroon" title="c20">H</font> as   <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"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green" title="E41">A4</font></i></a></i>;<br/>
<a NAME="E36:40"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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> holds  <font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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>;<br/>

<b>consider </b><font color="Maroon" title="c23">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <font color="Maroon" title="c24">x2</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> <b> such that </b><br/><a NAME="E2:40_5"/><i><font color="Green" title="E58">A38</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c23">x1</font>,<font color="Maroon" title="c24">x2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c23">x1</font>,<font color="Maroon" title="c24">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a>, <a class="txt" href="topalg_5.html#E2:40_5"><i><font color="Green" title="E58">A38</font></i></a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:40"><i><font color="Green" title="E57">A37</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">f1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x2</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c14">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c23">x1</font>,<font color="Maroon" title="c24">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E31:40"><i><font color="Green" title="E55">A35</font></i></a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green" title="E40">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c14">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5"><i><font color="Green" title="E58">A38</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E37:40"/><b>then </b><i><font color="Green" title="E58">A39</font></i>: 
<font color="Maroon" title="c14">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">H</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<a NAME="E38:40"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>   st <font color="Maroon" title="c21">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> ex <font color="Olive" title="b3">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"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c21">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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>;<br/><b>let </b><font color="Maroon" title="c23">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40_6"/><i><font color="Green" title="E59">A40</font></i>: 
<font color="Maroon" title="c21">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">V</font>
 <b>and </b><br/><a NAME="E2:40_6"/><i><font color="Green" title="E60">A41</font></i>: 
<font color="Maroon" title="c23">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c24">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <font color="Maroon" title="c25">p2</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> <b> such that </b><br/><a NAME="E4:40_6"/><i><font color="Green" title="E61">A42</font></i>: 
<font color="Maroon" title="c22">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c24">p1</font>,<font color="Maroon" title="c25">p2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<font color="Maroon" title="c21">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">p</font> = 
<font color="Maroon" title="c21">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c24">p1</font>,<font color="Maroon" title="c25">p2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_6"><i><font color="Green" title="E61">A42</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">p2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:40"><i><font color="Green" title="E57">A37</font></i></a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c26">W</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="E7:40_6"/><i><font color="Green" title="E62">A43</font></i>: 
<font color="Maroon" title="c25">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">W</font>
 <b>and </b><br/><a NAME="E8:40_6"/><i><font color="Green" title="E63">A44</font></i>: 
<font color="Maroon" title="c26">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E9:40_6"/><i><font color="Green" title="E64">A45</font></i>: 
<font color="Maroon" title="c19">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c26">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_6"><i><font color="Green" title="E59">A40</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_6"><i><font color="Green" title="E60">A41</font></i></a>, <a class="txt" href="topalg_5.html#E30:40"><i><font color="Green" title="E54">A34</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>take </b><font color="Maroon" title="c27">W1</font> = <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c26">W</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>;<br/>

<b>thus </b><a NAME="E10:40_6"/>
<font color="Maroon" title="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">W1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_6"><i><font color="Green" title="E61">A42</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_6"><i><font color="Green" title="E62">A43</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="E11:40_6"/>
<font color="Maroon" title="c27">W1</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#E8:40_6"><i><font color="Green" title="E63">A44</font></i></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon" title="c28">y</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="E12:40_6"/>
<font color="Maroon" title="c28">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c27">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c29">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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="E14:40_6"/><i><font color="Green" title="E65">A46</font></i>: 
<font color="Maroon" title="c29">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">W1</font>
 <b>and </b><br/><a NAME="E15:40_6"/><i><font color="Green" title="E66">A47</font></i>: 
<font color="Maroon" title="c28">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">c</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c30">c1</font>, <font color="Maroon" title="c31">c2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:40_6"/><i><font color="Green" title="E67">A48</font></i>: 
<font color="Maroon" title="c30">c1</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="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E18:40_6"/><i><font color="Green" title="E68">A49</font></i>: 
<font color="Maroon" title="c31">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">W</font>
 <b>and </b><br/><a NAME="E19:40_6"/><i><font color="Green" title="E69">A50</font></i>: 
<font color="Maroon" title="c29">c</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="c30">c1</font>,<font color="Maroon" title="c31">c2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:40_6"><i><font color="Green" title="E65">A46</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<i><font color="Green" title="E70">A51</font></i>: <font color="Maroon" title="c21">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">c</font> = 
<font color="Maroon" title="c21">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">c1</font>,<font color="Maroon" title="c31">c2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E19:40_6"><i><font color="Green" title="E69">A50</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c31">c2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:40"><i><font color="Green" title="E57">A37</font></i></a>, <a class="txt" href="topalg_5.html#E17:40_6"><i><font color="Green" title="E67">A48</font></i></a>, <a class="txt" href="topalg_5.html#E18:40_6"><i><font color="Green" title="E68">A49</font></i></a></i>
;<br/>
<a NAME="E21:40_6"/>
<font color="Maroon" title="c19">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c31">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">f1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c26">W</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E18:40_6"><i><font color="Green" title="E68">A49</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E22:40_6"/>
<font color="Maroon" title="c28">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:40_6"><i><font color="Green" title="E64">A45</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_6"><i><font color="Green" title="E66">A47</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_6"><i><font color="Green" title="E70">A51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:40"/><b>then </b><i><font color="Green" title="E59">A52</font></i>: 
<font color="Maroon" title="c21">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E40:40"/>
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>  holds  <font color="Maroon" title="c18">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">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/>

<a NAME="E1:40_7"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">H</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 <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:40_7"/><b>then </b><i><font color="Green" title="E60">A53</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><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 <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green" title="E43">A17</font></i></a>, <a class="txt" href="topalg_5.html#E12:40"><i><font color="Green" title="E44">A18</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E3:40_7"/><i><font color="Green" title="E61">A54</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="txt" href="topalg_5.html#E14:40"><i><font color="Green" title="E46">A20</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="E4:40_7"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><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 <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c21">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><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 <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_7_1"/><i><font color="Green" title="E62">A55</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><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 <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">x1</font>, <font color="Maroon" title="c25">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_7_1"/><i><font color="Green" title="E63">A56</font></i>: 
<font color="Maroon" title="c24">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_7_1"/><i><font color="Green" title="E64">A57</font></i>: 
<font color="Maroon" title="c25">x2</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">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E5:40_7_1"/><i><font color="Green" title="E65">A58</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">x1</font>,<font color="Maroon" title="c25">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_7"><i><font color="Green" title="E60">A53</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:40_7_1"/><i><font color="Green" title="E66">A59</font></i>: 
<font color="Maroon" title="c25">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j0</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_7_1"><i><font color="Green" title="E64">A57</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c21">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><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 <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c23">x</font> = 
<font color="Maroon" title="c21">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c24">x1</font>,<font color="Maroon" title="c25">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40_7_1"><i><font color="Green" title="E65">A58</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_7_1"><i><font color="Green" title="E62">A55</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c19">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c25">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:40"><i><font color="Green" title="E57">A37</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_7_1"><i><font color="Green" title="E63">A56</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_7_1"><i><font color="Green" title="E66">A59</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p3">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c23">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_7_1"><i><font color="Green" title="E63">A56</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_7_1"><i><font color="Green" title="E64">A57</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_7_1"><i><font color="Green" title="E65">A58</font></i></a>, <a class="txt" href="topalg_5.html#E16:40"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_7_1"><i><font color="Green" title="E66">A59</font></i></a>, <a class="txt" href="topalg_5.html#E32:40"><i><font color="Green" title="E56">A36</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:40_7"/><b>then </b><i><font color="Green" title="E62">A60</font></i>: 
<font color="Maroon" title="c21">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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 <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c10">xt</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_7"><i><font color="Green" title="E60">A53</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_7"><i><font color="Green" title="E61">A54</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c18">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font> = 
<font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c22">i</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c21">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c22">i</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E39:40"><i><font color="Green" title="E59">A52</font></i></a>, <a class="txt" href="topalg_5.html#E37:40"><i><font color="Green" title="E58">A39</font></i></a>, <a class="txt" href="topalg_5.html#E23:40"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_7"><i><font color="Green" title="E62">A60</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:40"><i><font color="Green" title="E57">A37</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E41:40"/>
<font color="Maroon" title="c18">ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c19">f1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
