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<a NAME="E1:45_1_1"/><i><font color="Green" title="E41">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:45_1_1"/>
 <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c9">n</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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_1_1_1"/><i><font color="Green" title="E42">A2</font></i>: 
( <font color="Maroon" title="c8">m</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> <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  &amp; <font color="Maroon" title="c9">n</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> <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  )
 <b>and </b><br/><a NAME="E2:45_1_1_1"/><i><font color="Green" title="E43">A3</font></i>: 
<a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">m</font> = <font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c9">n</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="txt" href="topalg_5.html#E1:45_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<a NAME="E4:45_1_1_1"/>
( <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">m</font></span>)</span> &amp; <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c11">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="topalg_5.html#D5" target="_self" title="TOPALG_5:def.5">Def5</a></i>;<br/>
<a NAME="E5:45_1_1_1"/><b>then </b><i><font color="Green" title="E44">A4</font></i>: 
 <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">m</font>, <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c11">n</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:45_1_1_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="topalg_1.html#T47" title="TOPALG_1:th.47">TOPALG_1:47</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">F</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"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<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><b> such that </b><br/><a NAME="E7:45_1_1_1"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c12">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E8:45_1_1_1"/><i><font color="Green" title="E46">A6</font></i>: 
for <font color="Olive" title="b1">s</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 <br/> ( <font color="Maroon" title="c12">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c12">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; ( for <font color="Olive" title="b2">t</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 <br/> ( <font color="Maroon" title="c12">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a>  ) ) )
 <b>by </b><i><a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a></i>;<br/>
<a NAME="E9:45_1_1_1"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c12">F</font> is    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of  <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">m</font>, <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c11">n</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:45_1_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="topalg_5.html#E7:45_1_1_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="topalg_5.html#E8:45_1_1_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">xt</font> =  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E10:45_1_1_1"/><i><font color="Green" title="E48">A8</font></i>: 
 <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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"><a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="toprealb.html#T34" title="TOPREALB:th.34">TOPREALB:34</a>, <a class="ref" href="toprealb.html#D2" title="TOPREALB:def.2">TOPREALB:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">yt</font> being   <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> , <font color="Maroon" title="c15">Pt</font>, <font color="Maroon" title="c16">Qt</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c14">yt</font>, <font color="Maroon" title="c17">Ft</font> being    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c15">Pt</font>,<font color="Maroon" title="c16">Qt</font><b> such that </b><br/><a NAME="E12:45_1_1_1"/><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c15">Pt</font>,<font color="Maroon" title="c16">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> 
 <b>and </b><br/><a NAME="E13:45_1_1_1"/><i><font color="Green" title="E50">A10</font></i>: 
<font color="Maroon" title="c12">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">Ft</font>
 <b>and </b><br/><a NAME="E14:45_1_1_1"/>
<font color="Maroon" title="c14">yt</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"><a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E15:45_1_1_1"/>
for <font color="Olive" title="b1">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c15">Pt</font>,<font color="Maroon" title="c16">Qt</font>  st <font color="Maroon" title="c12">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">F1</font> holds <br/><font color="Maroon" title="c17">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:45_1_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="topalg_5.html#E9:45_1_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="topalg_5.html#T24" target="_self" title="TOPALG_5:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">ftm</font> being   <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> <b> such that </b><br/><a NAME="E17:45_1_1_1"/>
<font color="Maroon" title="c18">ftm</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E18:45_1_1_1"/>
 <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">m</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">ftm</font>
 <b>and </b><br/><a NAME="E19:45_1_1_1"/>
<font color="Maroon" title="c18">ftm</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E20:45_1_1_1"/><i><font color="Green" title="E51">A11</font></i>: 
for <font color="Olive" title="b1">f1</font> being  <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>   st <font color="Olive" title="b1">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp;  <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">m</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">f1</font> &amp; <font color="Olive" title="b1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c18">ftm</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:45_1_1_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="topalg_5.html#T23" target="_self" title="TOPALG_5:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">ftn</font> being   <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> <b> such that </b><br/><a NAME="E22:45_1_1_1"/>
<font color="Maroon" title="c19">ftn</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E23:45_1_1_1"/>
 <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c11">n</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">ftn</font>
 <b>and </b><br/><a NAME="E24:45_1_1_1"/>
<font color="Maroon" title="c19">ftn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E25:45_1_1_1"/><i><font color="Green" title="E52">A12</font></i>: 
for <font color="Olive" title="b1">f1</font> being  <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>   st <font color="Olive" title="b1">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp;  <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c11">n</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">f1</font> &amp; <font color="Olive" title="b1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c19">ftn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:45_1_1_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="topalg_5.html#T23" target="_self" title="TOPALG_5:th.23">Th23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">ft0</font> =  <a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">Ft</font>;<br/>
<b>set </b><font color="Maroon" title="c21">ft1</font> =  <a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">Ft</font>;<br/>
<i><font color="Green" title="E53">A13</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c10">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j0</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D1" target="_self" title="TOPALG_5:def.1">Def1</a></i>
<br/>.= 
 <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="toprealb.html#D2" title="TOPREALB:def.2">TOPREALB:def 2</a></i>
;<br/>
<i><font color="Green" title="E54">A14</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">Ft</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c17">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c4">j0</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</a></i>
<br/>.= 
 <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:45_1_1_1"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
;<br/>
<a NAME="E28:45_1_1_1"/>
 <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">m</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> <span class="p1">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c10">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T20" target="_self" title="TOPALG_5:th.20">Th20</a></i>;<br/>
<a NAME="E29:45_1_1_1"/><b>then </b><i><font color="Green" title="E55">A15</font></i>: 
 <a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c10">m</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">ftm</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:45_1_1_1"><i><font color="Green" title="E51">A11</font></i></a>, <a class="txt" href="topalg_5.html#E26:45_1_1_1"><i><font color="Green" title="E53">A13</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_3"><b>now </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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<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> <span class="p2">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">Ft</font></span>)</span></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_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">Ft</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</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">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c4">j0</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c12">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c4">j0</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:45_1_1_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green" title="E5">Lm5</font></i></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">m</font></span>)</span> <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#E8:45_1_1_1"><i><font color="Green" title="E46">A6</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E31:45_1_1_1"/><b>then </b>
 <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">m</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> <span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">Ft</font></span>)</span>
 
<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="E32:45_1_1_1"/><b>then </b>
 <a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">ftm</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:45_1_1_1"><i><font color="Green" title="E51">A11</font></i></a>, <a class="txt" href="topalg_5.html#E27:45_1_1_1"><i><font color="Green" title="E54">A14</font></i></a></i>;<br/>
<a NAME="E33:45_1_1_1"/><b>then </b><i><font color="Green" title="E56">A16</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">Ft</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1
 
<b>by </b><i><a class="txt" href="topalg_5.html#E29:45_1_1_1"><i><font color="Green" title="E55">A15</font></i></a>, <a class="ref" href="topalg_5.html#D1" target="_self" title="TOPALG_5:def.1">Def1</a></i>;<br/>
<i><font color="Green" title="E57">A17</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c11">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j0</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D1" target="_self" title="TOPALG_5:def.1">Def1</a></i>
<br/>.= 
 <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="toprealb.html#D2" title="TOPREALB:def.2">TOPREALB:def 2</a></i>
;<br/>
<a NAME="E35:45_1_1_1"/>
 <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c11">n</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> <span class="p1">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c11">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T20" target="_self" title="TOPALG_5:th.20">Th20</a></i>;<br/>
<a NAME="E36:45_1_1_1"/><b>then </b><i><font color="Green" title="E58">A18</font></i>: 
 <a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c11">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c19">ftn</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E25:45_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="topalg_5.html#E34:45_1_1_1"><i><font color="Green" title="E57">A17</font></i></a></i>;<br/>
<i><font color="Green" title="E59">A19</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">Ft</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c17">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">j1</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</a></i>
<br/>.= 
 <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:45_1_1_1"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_6"><b>now </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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<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> <span class="p2">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">Ft</font></span>)</span></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_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">Ft</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</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">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c5">j1</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c12">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c5">j1</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:45_1_1_1"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green" title="E5">Lm5</font></i></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c11">n</font></span>)</span> <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#E8:45_1_1_1"><i><font color="Green" title="E46">A6</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E39:45_1_1_1"/><b>then </b>
 <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c11">n</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> <span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">Ft</font></span>)</span>
 
<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="E40:45_1_1_1"/><b>then </b>
 <a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c19">ftn</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E25:45_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="topalg_5.html#E37:45_1_1_1"><i><font color="Green" title="E59">A19</font></i></a></i>;<br/>
<a NAME="E41:45_1_1_1"/><b>then </b><i><font color="Green" title="E60">A20</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">Ft</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1
 
<b>by </b><i><a class="txt" href="topalg_5.html#E36:45_1_1_1"><i><font color="Green" title="E58">A18</font></i></a>, <a class="ref" href="topalg_5.html#D1" target="_self" title="TOPALG_5:def.1">Def1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">Ft</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font> = 
<font color="Maroon" title="c17">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c4">j0</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c14">yt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E12:45_1_1_1"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c17">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c5">j1</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E12:45_1_1_1"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">Ft</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E43:45_1_1_1"/>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E33:45_1_1_1"><i><font color="Green" title="E56">A16</font></i></a>, <a class="txt" href="topalg_5.html#E41:45_1_1_1"><i><font color="Green" title="E60">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:45_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c8">q</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="E4:45_1_1"/>
<font color="Maroon" title="c8">q</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="topalg_1.html#NK4" title="TOPALG_1:NK.4">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">f</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of  <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> <b> such that </b><br/><a NAME="E6:45_1_1"/><i><font color="Green" title="E42">A21</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>,<font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" title="TOPALG_1:th.48">TOPALG_1:48</a></i>;<br/>
<a NAME="E7:45_1_1"/>
 <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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"><a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="toprealb.html#T34" title="TOPREALB:th.34">TOPREALB:34</a>, <a class="ref" href="toprealb.html#D2" title="TOPREALB:def.2">TOPREALB:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">ft</font> being   <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> <b> such that </b><br/><a NAME="E9:45_1_1"/><i><font color="Green" title="E43">A22</font></i>: 
<font color="Maroon" title="c10">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E10:45_1_1"/><i><font color="Green" title="E44">A23</font></i>: 
<font color="Maroon" title="c9">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="c10">ft</font>
 <b>and </b><br/><a NAME="E11:45_1_1"/><i><font color="Green" title="E45">A24</font></i>: 
<font color="Maroon" title="c10">ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E12:45_1_1"/>
for <font color="Olive" title="b1">f1</font> being  <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>   st <font color="Olive" title="b1">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">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">f1</font> &amp; <font color="Olive" title="b1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c10">ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f1</font>
 <b>by </b><i><a class="ref" href="topalg_5.html#T23" target="_self" title="TOPALG_5:th.23">Th23</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="c10">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font></span>)</span> = 
<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="c10">ft</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font>
<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#K9" title="TOPREALB:func.9">c[10]</a> 
<b>by </b><i><a class="txt" href="topalg_5.html#E10:45_1_1"><i><font color="Green" title="E44">A23</font></i></a>, <a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>
;<br/>
<a NAME="E14:45_1_1"/><b>then </b>
<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="c10">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span> <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"><a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:45_1_1"/><b>then </b><i><font color="Green" title="E46">A25</font></i>: 
<font color="Maroon" title="c10">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="toprealb.html#T34" title="TOPREALB:th.34">TOPREALB:34</a></i>;<br/>
<a NAME="E16:45_1_1"/><b>then </b><i><font color="Green" title="E47">A26</font></i>: 
<font color="Maroon" title="c10">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 
<b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<a NAME="E17:45_1_1"/>
<font color="Maroon" title="c10">ft</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:45_1_1_3"/>
<font color="Maroon" title="c10">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#E11:45_1_1"><i><font color="Green" title="E45">A24</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a><br/>

<b>thus </b><a NAME="E2:45_1_1_3"/>
<font color="Maroon" title="c10">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:45_1_1"><i><font color="Green" title="E43">A22</font></i></a></i>;<br/>

<b>thus </b><a NAME="E3:45_1_1_3"/>
<font color="Maroon" title="c10">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="toprealb.html#D2" title="TOPREALB:def.2">TOPREALB:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:45_1_1"/><b>then </b>
<a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" title="TOPALG_1:func.2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>,<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="c10">ft</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E16:45_1_1"><i><font color="Green" title="E47">A26</font></i></a>, <a class="ref" href="topalg_5.html#T25" target="_self" title="TOPALG_5:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E19:45_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="topalg_5.html#K8" title="TOPALG_5:func.8">Ciso</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:45_1_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="topalg_5.html#E6:45_1_1"><i><font color="Green" title="E42">A21</font></i></a>, <a class="txt" href="topalg_5.html#E10:45_1_1"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="topalg_5.html#E15:45_1_1"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div>
