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<span class="kw">set </span><font color="Maroon" title="c8">f</font> =  <a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="group_6.html#D6" title="GROUP_6:def.6">GROUP_6:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">fX</font>, <font color="Maroon" title="c12">fY</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> <span class="kw"> such that </span><br/><a NAME="E2:42_1_1"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</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#K4" title="TOPALG_1:func.4">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="c11">fX</font>)
 <span class="kw">and </span><br/><a NAME="E3:42_1_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</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#K4" title="TOPALG_1:func.4">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="c12">fY</font>)
 <span class="kw">and </span><br/><a NAME="E4:42_1_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">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> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></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#K4" title="TOPALG_1:func.4">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"><font color="Maroon" title="c11">fX</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c12">fY</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_1.html#D5" title="TOPALG_1:def.5">TOPALG_1:def 5</a></span>;<br/>
<a NAME="E5:42_1_1"/>
<a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</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#K4" title="TOPALG_1:func.4">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">y</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D5" target="_self" title="TOPALG_5:def.5">Def5</a></span>;<br/>
<a NAME="E6:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c12">fY</font>, <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">y</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="topalg_1.html#T46" title="TOPALG_1:th.46">TOPALG_1:46</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">tx</font> =  <a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c9">x</font>) as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c13">tx</font> <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">y</font></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E45">A5</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c13">tx</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<font color="Maroon" title="c13">tx</font> <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#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">j0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c13">tx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D1" target="_self" title="TOPALG_5:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#D4" title="FCONT_1:def.4">FCONT_1:def 4</a></span>
<br/>.= 
 <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D2" title="TOPREALB:def.2">TOPREALB:def 2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E46">A6</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c13">tx</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon" title="c13">tx</font> <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#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c13">tx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">j1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D1" target="_self" title="TOPALG_5:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#D4" title="FCONT_1:def.4">FCONT_1:def 4</a></span>
<br/>.= 
 <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D2" title="TOPREALB:def.2">TOPREALB:def 2</a></span>
;<br/>
<a NAME="E10:42_1_1"/>
<font color="Maroon" title="c9">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E11:42_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">tx</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan16.html#T20" title="JORDAN16:th.20">JORDAN16:20</a></span>;<br/>
<a NAME="E12:42_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">tx</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">p</font> = <font color="Maroon" title="c13">tx</font> <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">y</font></span>)</span> as    <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> <font color="Maroon" title="c9">x</font>, <a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_1_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E9:42_1_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></span>;<br/>
<a NAME="E14:42_1_1"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 for <font color="Olive" title="b1">a</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  <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">a</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1_3_1"/>
( <font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 or 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_3_1_1"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font></span><br/><div class="add"><a NAME="E2:42_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A9</font></span>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font> is   <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T3" title="BORSUK_6:th.3">BORSUK_6:3</a></span>;<br/><span class="kw">thus </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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</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"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_3_1_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="borsuk_6.html#D2" title="BORSUK_6:def.2">BORSUK_6:def 2</a></span>
<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="c9">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_3_1_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="topalg_5.html#D1" target="_self" title="TOPALG_5:def.1">Def1</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D11" title="TOPREALB:def.11">TOPREALB:def 11</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_3_1_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="topalg_5.html#D4" target="_self" title="TOPALG_5:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_3_1_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="borsuk_6.html#D2" title="BORSUK_6:def.2">BORSUK_6:def 2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_3_1_2"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font></span><br/><div class="add"><a NAME="E2:42_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 is   <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T4" title="BORSUK_6:th.4">BORSUK_6:4</a></span>;<br/><span class="kw">thus </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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</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"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<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="c15">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_3_1_2"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="ref" href="borsuk_6.html#D2" title="BORSUK_6:def.2">BORSUK_6:def 2</a></span>
<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="c13">tx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_3_1_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<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="c13">tx</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_3_1_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="topalg_5.html#D1" target="_self" title="TOPALG_5:def.1">Def1</a></span>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#D4" title="FCONT_1:def.4">FCONT_1:def 4</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D11" title="TOPREALB:def.11">TOPREALB:def 11</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T9" title="COMPLEX2:th.9">COMPLEX2:9</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T8" title="COMPLEX2:th.8">COMPLEX2:8</a></span>
<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">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_3_1_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="topalg_5.html#D4" target="_self" title="TOPALG_5:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_3_1_2"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="ref" href="borsuk_6.html#D2" title="BORSUK_6:def.2">BORSUK_6:def 2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E15:42_1_1"/>
<a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</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#K4" title="TOPALG_1:func.4">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="c9">x</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D5" target="_self" title="TOPALG_5:def.5">Def5</a></span>;<br/>
<a NAME="E16:42_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">fX</font>, <a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="topalg_1.html#T46" title="TOPALG_1:th.46">TOPALG_1:46</a></span>;<br/>
<a NAME="E17:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A12</font></span>: 
<font color="Maroon" title="c11">fX</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c12">fY</font>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.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">y</font></span>)</span> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="borsuk_6.html#T76" title="BORSUK_6:th.76">BORSUK_6:76</a></span>;<br/>
<span class="kw">thus </span><a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y</font></span>)</span> = 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" title="TOPALG_5:func.2">ExtendInt</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#T25" target="_self" title="TOPALG_5:th.25">Th25</a></span>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">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"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" title="TOPALG_5:func.6">cLoop</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K7" title="TOPALG_5:func.7">Ciso</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E17:42_1_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="topalg_1.html#T46" title="TOPALG_1:th.46">TOPALG_1:46</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
