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

<b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>, <font color="Maroon">y0</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">x0</font>,<font color="Maroon">y0</font>, <font color="Maroon">Q</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">x0</font>,<font color="Maroon">y0</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">P</font>,<font color="Maroon">Q</font>;<br/><b>let </b><font color="Maroon">xt</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>







<b>assume </b><a NAME="E1:42"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#R4">are_homotopic</a> 
 ;<br/>

<b>assume </b><a NAME="E2:42"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">xt</font> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">ft</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E4:42"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">ft</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">xt</font>
 <b>and </b><br/><a NAME="E5:42"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">P</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ft</font>
 <b>and </b><br/><a NAME="E6:42"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">ft</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E7:42"/>
for <font color="Olive">f1</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">f1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">P</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">f1</font> &amp; <font color="Olive">f1</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">xt</font> holds <br/><font color="Maroon">ft</font> <a href="funct_2.html#R2">=</a> <font color="Olive">f1</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:42"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>;<br/>
<a NAME="E9:42"/>
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">xt</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E10:42"/><b>then </b><i><font color="Green">E81</font></i>: 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">xt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E11:42"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
;<br/>
<b>consider </b><font color="Maroon">Ft</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E13:42"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">i</font>,<font color="Maroon">Ft</font> <a href="metric_1.html#K1">.</a> <font color="Olive">y</font>,<font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<a NAME="E14:42"/><i><font color="Green">E85</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:42"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E16:42"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E17:42"/><i><font color="Green">E88</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Ft</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:42"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Ft</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E19:42"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Ft</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Ft</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E20:42"/><i><font color="Green">E90</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Ft</font> = <font color="Maroon">Ft</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E22:42"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Maroon">Ft</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Ft</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:42_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">Ft</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:42_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span><b> such that </b><br/><a NAME="E4:42_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E5:42_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">Ft</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>]
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">W1</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E7:42_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E8:42_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E9:42_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">ft</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">W</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>

<b>thus </b><a NAME="E11:42_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<b>thus </b><a NAME="E12:42_1"/>
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>

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

<b>assume </b><a NAME="E13:42_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ft</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E15:42_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E16:42_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">W1</font>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span><b> such that </b><br/><a NAME="E18:42_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E19:42_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">Ft</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:42_1"/>
<font color="Maroon">ft</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ft</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E21:42_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:42"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">Ft</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E24:42"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Ft</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Ft</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:42_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Ft</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>and </b><br/><a NAME="E4:42_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E5:42_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, , , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:42_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,0
<b>by </b><i>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, <a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ft</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, , <a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Ft</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E25:42"/><b>then </b>
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Ft</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E27:42"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E28:42"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E29:42"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font>
 <b>and </b><br/><a NAME="E30:42"/><i><font color="Green">E189</font></i>: 
for <font color="Olive">H</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#R2">=</a> <font color="Olive">H</font>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/>
<b>take </b><font color="Maroon">yt</font> = <font color="Maroon">ft</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font>;<br/>

<b>reconsider </b><font color="Maroon">ft</font> = <font color="Maroon">ft</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">xt</font>,<font color="Maroon">yt</font> <b>by </b><i>, , <a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">sM0</font> =  <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">sM1</font> =  <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font>;<br/>
<a NAME="E32:42"/><i><font color="Green">E190</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">j0</font>,<font color="Maroon">j0</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E33:42"/><i><font color="Green">E191</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">j1</font>,<font color="Maroon">j0</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green">E192</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j0</font>,<font color="Maroon">j0</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">Ft</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j0</font>,<font color="Maroon">j0</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">xt</font>
<b>by </b><i>, , </i>
;<br/>
<a NAME="E35:42"/>
for <font color="Olive">H</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  holds   <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a>
 
;<br/>
<a NAME="E36:42"/><b>then </b><i><font color="Green">E193</font></i>: 
 <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<i><font color="Green">E194</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j1</font>,<font color="Maroon">j0</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">Ft</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j1</font>,<font color="Maroon">j0</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">yt</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E38:42"/>
for <font color="Olive">H</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  holds   <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a>
 
;<br/>
<a NAME="E39:42"/><b>then </b><i><font color="Green">E195</font></i>: 
 <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">cP1</font> being  <a href="funct_1.html#V3">constant</a> <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">x0</font>;<br/>
<b>consider </b><font color="Maroon">ft1</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E42:42"/>
( <font color="Maroon">ft1</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">xt</font> &amp; <font color="Maroon">cP1</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ft1</font> &amp; <font color="Maroon">ft1</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>and </b><br/><a NAME="E43:42"/><i><font color="Green">E197</font></i>: 
for <font color="Olive">f1</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">f1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">cP1</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">f1</font> &amp; <font color="Olive">f1</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">xt</font> holds <br/><font color="Maroon">ft1</font> <a href="funct_2.html#R2">=</a> <font color="Olive">f1</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E44:42"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <font color="Maroon">cP1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j0</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j0</font>,<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="binop_1.html#T30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon">x0</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, , <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon">cP1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="topalg_3.html#T22">TOPALG_3:22</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E45:42"/><b>then </b>
<font color="Maroon">cP1</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E46:42"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">ft1</font> <a href="funct_2.html#R2">=</a>  <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">g1</font> = <a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">xt</font>;<br/>
<a NAME="E47:42"/><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">xt</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xt</font>
 
<b>by </b><i><a class="ref" href="topalg_3.html#T5">TOPALG_3:5</a></i>;<br/>
<a NAME="E48:42"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <font color="Maroon">cP1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">xt</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>thus </b><font color="Maroon">cP1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="topalg_3.html#T22">TOPALG_3:22</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">xt</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topalg_3.html#T5">TOPALG_3:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">xt</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E49:42"/><b>then </b>
<font color="Maroon">cP1</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">xt</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E50:42"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">ft1</font> <a href="funct_2.html#R2">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">xt</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">cP2</font> being  <a href="funct_1.html#V3">constant</a> <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">y0</font>;<br/>
<i><font color="Green">E201</font></i>: <a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">yt</font> = 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">y0</font>
<b>by </b><i><a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>
;<br/>
<a NAME="E53:42"/><b>then </b><i><font color="Green">E202</font></i>: 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">yt</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y0</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E54:42"/><b>then </b>
<font color="Maroon">yt</font> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y0</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">ft2</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E56:42"/>
( <font color="Maroon">ft2</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">yt</font> &amp; <font color="Maroon">cP2</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">ft2</font> &amp; <font color="Maroon">ft2</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>and </b><br/><a NAME="E57:42"/><i><font color="Green">E203</font></i>: 
for <font color="Olive">f1</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">f1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">cP2</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">f1</font> &amp; <font color="Olive">f1</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">yt</font> holds <br/><font color="Maroon">ft2</font> <a href="funct_2.html#R2">=</a> <font color="Olive">f1</font>
 <b>by </b><i/>;<br/>
<a NAME="E58:42"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <font color="Maroon">cP2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j1</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j1</font>,<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="binop_1.html#T30">BINOP_1:30</a></i>
<br/>.= 
<font color="Maroon">y0</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, , <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon">cP2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="topalg_3.html#T22">TOPALG_3:22</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E59:42"/><b>then </b>
<font color="Maroon">cP2</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E60:42"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">ft2</font> <a href="funct_2.html#R2">=</a>  <a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="topalg_5.html#E1:42"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">g2</font> = <a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">yt</font>;<br/>
<a NAME="E61:42"/><i><font color="Green">E205</font></i>: 
<span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">yt</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yt</font>
 
<b>by </b><i><a class="ref" href="topalg_3.html#T5">TOPALG_3:5</a></i>;<br/>
<a NAME="E62:42"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <font color="Maroon">cP2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">yt</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>thus </b><font color="Maroon">cP2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">y0</font>
<b>by </b><i><a class="ref" href="topalg_3.html#T22">TOPALG_3:22</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">yt</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topalg_3.html#T5">TOPALG_3:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">yt</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E63:42"/><b>then </b>
<font color="Maroon">cP2</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">yt</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E64:42"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">ft2</font> <a href="funct_2.html#R2">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">yt</font>
 
<b>by </b><i>, <a class="txt" href="topalg_5.html#E1:42"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">Qt</font> =  <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font>;<br/>
<a NAME="E65:42"/>
 <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">xt</font>,<font color="Maroon">yt</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_10"><b>proof </b></a><div class="add">

<a NAME="E1:42_10"/>
for <font color="Olive">H</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  holds   <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">j1</font>,<font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a>
 
;<br/>
<b>hence </b><a NAME="E2:42_10"/>
 <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j0</font>,<font color="Maroon">j1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">xt</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T1" target="_self">Th1</a>, , <a class="ref" href="topalg_3.html#T5">TOPALG_3:5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 = 
<font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j1</font>,<font color="Maroon">j1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">yt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:42"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E2:42"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topalg_3.html#T5">TOPALG_3:5</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Qt</font> =  <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">xt</font>,<font color="Maroon">yt</font> ;<br/>
<b>take </b>
<font color="Maroon">ft</font>
;<br/>

<b>take </b>
<font color="Maroon">Qt</font>
;<br/>

<div><i><font color="Green">E209</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><a NAME="E1:42_11"/><i><font color="Green">E210</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,0</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,0 = 
<font color="Maroon">Ft</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">j0</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">ft</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
<b>by </b><i>, </i>
;<br/><b>thus </b><a NAME="E3:42_11"/>
<font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">Qt</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>thus </b><font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j0</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">xt</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T1" target="_self">Th1</a>, , <a class="ref" href="topalg_3.html#T5">TOPALG_3:5</a></i>
;<br/><b>thus </b><font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" target="_self">Prj2</a> <font color="Maroon">j1</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">yt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:42"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E2:42"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topalg_3.html#T5">TOPALG_3:5</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E68:42"/>
<font color="Maroon">G</font> is    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">ft</font>,<font color="Maroon">Qt</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_12"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_12"/>
<font color="Maroon">ft</font>,<font color="Maroon">Qt</font> <a href="borsuk_2.html#R4">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, , <a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a><br/>

<b>thus </b><a NAME="E2:42_12"/>
( <font color="Maroon">G</font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/> ( <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">ft</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">Qt</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/> ( <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">xt</font> &amp; <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">yt</font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">ft</font>,<font color="Maroon">Qt</font> ;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E70:42"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">ft</font>,<font color="Maroon">Qt</font> <a href="borsuk_2.html#R4">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, , <a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a></i>;<br/>

<b>thus </b><a NAME="E71:42"/>
<font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a></i>;<br/>

<b>thus </b><a NAME="E72:42"/>
<font color="Maroon">yt</font> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y0</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>

<b>let </b><font color="Maroon">F1</font> be    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">ft</font>,<font color="Maroon">Qt</font>;<br/>

<b>assume </b><a NAME="E73:42"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F1</font>
 ;<br/>

<a NAME="E74:42"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E75:42"/><b>then </b><i><font color="Green">E213</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E76:42"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:42_13"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_13"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>and </b><br/><a NAME="E4:42_13"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E5:42_13"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8:42"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E6:42"><i><font color="Green">E79</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:42_13"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">F1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:42"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E8:42"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E10:42"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E13:42"><i><font color="Green">E84</font></i></a>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon">Ft</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E10:42"><i><font color="Green">E81</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:42"><i><font color="Green">E83</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E77:42"/><b>then </b>
<font color="Maroon">F1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E6:42"><i><font color="Green">E79</font></i></a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E78:42"/>
<font color="Maroon">G</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:42"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a></i>;<br/>


</div>
