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<b>let </b><font color="Maroon" title="c8">x0</font>, <font color="Maroon" title="c9">y0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c10">P</font>, <font color="Maroon" title="c11">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font>;<br/><b>let </b><font color="Maroon" title="c12">F</font> be    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font>;<br/><b>let </b><font color="Maroon" title="c13">xt</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/>







<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> 
 ;<br/>

<b>assume </b><a NAME="E2:41"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c13">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">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="E4:41"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c14">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font>
 <b>and </b><br/><a NAME="E5:41"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c10">P</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="c14">ft</font>
 <b>and </b><br/><a NAME="E6:41"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c14">ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E7:41"/>
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="c10">P</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> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font> holds <br/><font color="Maroon" title="c14">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 NAME="E8:41"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c12">F</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#E1:41"><i><font color="Green" title="E39">A1</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="E9:41"/>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:41"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<a NAME="E10:41"/><b>then </b><i><font color="Green" title="E45">A7</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">xt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1;<br/>
<a NAME="E11:41"/><i><font color="Green" title="E46">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>  ex <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c15">Ft</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E13:41"/><i><font color="Green" title="E47">A9</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c15">Ft</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">i</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="topalg_5.html#E11:41"><i><font color="Green" title="E46">A8</font></i></a>);<br/></i>
<a NAME="E14:41"/><i><font color="Green" title="E48">A10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:41"/><b>then </b><i><font color="Green" title="E49">A11</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">F</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5"><i><font color="Green" title="E5">Lm5</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E16:41"/><b>then </b><i><font color="Green" title="E50">A12</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E17:41"/><i><font color="Green" title="E51">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">Ft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:41"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">Ft</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<a NAME="E19:41"/><b>then </b><i><font color="Green" title="E52">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">Ft</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">Ft</font>
 
<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="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E20:41"/><i><font color="Green" title="E53">A15</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">Ft</font> = <font color="Maroon" title="c15">Ft</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E22:41"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>   st <font color="Maroon" title="c16">Ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c16">Ft</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/><b>let </b><font color="Maroon" title="c18">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:41_1"/><i><font color="Green" title="E54">A16</font></i>: 
<font color="Maroon" title="c16">Ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">V</font>
 <b>and </b><br/><a NAME="E2:41_1"/><i><font color="Green" title="E55">A17</font></i>: 
<font color="Maroon" title="c18">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c19">p1</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> , <font color="Maroon" title="c20">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span><b> such that </b><br/><a NAME="E4:41_1"/><i><font color="Green" title="E56">A18</font></i>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c20">p2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:41"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E5:41_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c20">p2</font>,<font color="Maroon" title="c16">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">p1</font>,<font color="Maroon" title="c20">p2</font>]
 
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="txt" href="topalg_5.html#E13:41"><i><font color="Green" title="E47">A9</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">W1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E7:41_1"/><i><font color="Green" title="E57">A19</font></i>: 
<font color="Maroon" title="c19">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">W1</font>
 <b>and </b><br/><a NAME="E8:41_1"/><i><font color="Green" title="E58">A20</font></i>: 
<font color="Maroon" title="c21">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E9:41_1"/><i><font color="Green" title="E59">A21</font></i>: 
<font color="Maroon" title="c14">ft</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="topalg_5.html#E1:41_1"><i><font color="Green" title="E54">A16</font></i></a>, <a class="txt" href="topalg_5.html#E2:41_1"><i><font color="Green" title="E55">A17</font></i></a>, <a class="txt" href="topalg_5.html#E4:41_1"><i><font color="Green" title="E56">A18</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">W1</font> = <font color="Maroon" title="c21">W1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E7:41_1"><i><font color="Green" title="E57">A19</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c23">W</font> = <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c22">W1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>;<br/>

<b>thus </b><a NAME="E11:41_1"/>
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">W</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:41_1"><i><font color="Green" title="E56">A18</font></i></a>, <a class="txt" href="topalg_5.html#E7:41_1"><i><font color="Green" title="E57">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>

<b>thus </b><a NAME="E12:41_1"/>
<font color="Maroon" title="c23">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8:41_1"><i><font color="Green" title="E58">A20</font></i></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>

<b>let </b><font color="Maroon" title="c24">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E13:41_1"/>
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Ft</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c23">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c25">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E15:41_1"/><i><font color="Green" title="E60">A22</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">W</font>
 <b>and </b><br/><a NAME="E16:41_1"/><i><font color="Green" title="E61">A23</font></i>: 
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">x1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c22">W1</font>, <font color="Maroon" title="c27">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span><b> such that </b><br/><a NAME="E18:41_1"/><i><font color="Green" title="E62">A24</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c26">x1</font>,<font color="Maroon" title="c27">x2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:41_1"><i><font color="Green" title="E60">A22</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E19:41_1"/><i><font color="Green" title="E63">A25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">x1</font>,<font color="Maroon" title="c27">x2</font>,<font color="Maroon" title="c16">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">x1</font>,<font color="Maroon" title="c27">x2</font>]
 
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="txt" href="topalg_5.html#E13:41"><i><font color="Green" title="E47">A9</font></i></a></i>;<br/>
<a NAME="E20:41_1"/>
<font color="Maroon" title="c14">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">ft</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c22">W1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E21:41_1"/>
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:41_1"><i><font color="Green" title="E59">A21</font></i></a>, <a class="txt" href="topalg_5.html#E16:41_1"><i><font color="Green" title="E61">A23</font></i></a>, <a class="txt" href="topalg_5.html#E18:41_1"><i><font color="Green" title="E62">A24</font></i></a>, <a class="txt" href="topalg_5.html#E19:41_1"><i><font color="Green" title="E63">A25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:41"/><b>then </b><i><font color="Green" title="E54">A26</font></i>: 
<font color="Maroon" title="c16">Ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E24:41"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">Ft</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">Ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_2"/><i><font color="Green" title="E55">A27</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">Ft</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c18">x1</font>, <font color="Maroon" title="c19">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:41_2"/><i><font color="Green" title="E56">A28</font></i>: 
<font color="Maroon" title="c18">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <b>and </b><br/><a NAME="E4:41_2"/><i><font color="Green" title="E57">A29</font></i>: 
<font color="Maroon" title="c19">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E5:41_2"/><i><font color="Green" title="E58">A30</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:41_2"><i><font color="Green" title="E55">A27</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="txt" href="topalg_5.html#E20:41"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:41_2"/>
<font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:41_2"><i><font color="Green" title="E57">A29</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> = 
<font color="Maroon" title="c12">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">x1</font>,0
<b>by </b><i><a class="txt" href="topalg_5.html#E5:41_2"><i><font color="Green" title="E58">A30</font></i></a>, <a class="txt" href="topalg_5.html#E1:41_2"><i><font color="Green" title="E55">A27</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="txt" href="topalg_5.html#E20:41"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x1</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:41"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="topalg_5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="topalg_5.html#E3:41_2"><i><font color="Green" title="E56">A28</font></i></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</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="c14">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:41_2"><i><font color="Green" title="E56">A28</font></i></a>, <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="c16">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">x1</font>,<font color="Maroon" title="c19">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:41"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="topalg_5.html#E3:41_2"><i><font color="Green" title="E56">A28</font></i></a>, <a class="txt" href="topalg_5.html#E4:41_2"><i><font color="Green" title="E57">A29</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">Ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:41_2"><i><font color="Green" title="E55">A27</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="txt" href="topalg_5.html#E5:41_2"><i><font color="Green" title="E58">A30</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E25:41"/><b>then </b>
<font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">Ft</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E16:41"><i><font color="Green" title="E50">A12</font></i></a>, <a class="txt" href="topalg_5.html#E19:41"><i><font color="Green" title="E52">A14</font></i></a>, <a class="txt" href="topalg_5.html#E17:41"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> <b> such that </b><br/><a NAME="E27:41"/><i><font color="Green" title="E55">A31</font></i>: 
<font color="Maroon" title="c17">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E28:41"/><i><font color="Green" title="E56">A32</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">G</font>
 <b>and </b><br/><a NAME="E29:41"/><i><font color="Green" title="E57">A33</font></i>: 
<font color="Maroon" title="c17">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Ft</font>
 <b>and </b><br/><a NAME="E30:41"/><i><font color="Green" title="E58">A34</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="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">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Ft</font> holds <br/><font color="Maroon" title="c17">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8:41"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="topalg_5.html#E23:41"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="topalg_5.html#T22" target="_self" title="TOPALG_5:th.22">Th22</a></i>;<br/>
<b>take </b><font color="Maroon" title="c18">yt</font> = <font color="Maroon" title="c14">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c19">ft</font> = <font color="Maroon" title="c14">ft</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Maroon" title="c18">yt</font> <b>by </b><i><a class="txt" href="topalg_5.html#E4:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="topalg_5.html#E6:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">sM0</font> =  <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</font>;<br/>
<b>set </b><font color="Maroon" title="c21">sM1</font> =  <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</font>;<br/>
<a NAME="E32:41"/><i><font color="Green" title="E59">A35</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c4">j0</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E33:41"/><i><font color="Green" title="E60">A36</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c4">j0</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green" title="E61">A37</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 = 
<font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c4">j0</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c16">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="txt" href="topalg_5.html#E32:41"><i><font color="Green" title="E59">A35</font></i></a>, <a class="txt" href="topalg_5.html#E29:41"><i><font color="Green" title="E57">A33</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c13">xt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="topalg_5.html#E13:41"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>
;<br/>
<a NAME="E35:41"/>
for <font color="Olive" title="b1">H</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  holds   <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
;<br/>
<a NAME="E36:41"/><b>then </b><i><font color="Green" title="E62">A38</font></i>: 
 <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</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#E27:41"><i><font color="Green" title="E55">A31</font></i></a></i>;<br/>
<i><font color="Green" title="E63">A39</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 = 
<font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c4">j0</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c16">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="txt" href="topalg_5.html#E33:41"><i><font color="Green" title="E60">A36</font></i></a>, <a class="txt" href="topalg_5.html#E29:41"><i><font color="Green" title="E57">A33</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c18">yt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:41"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>
;<br/>
<a NAME="E38:41"/>
for <font color="Olive" title="b1">H</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  holds   <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
;<br/>
<a NAME="E39:41"/><b>then </b><i><font color="Green" title="E64">A40</font></i>: 
 <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</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#E27:41"><i><font color="Green" title="E55">A31</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">cP1</font> being  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c8">x0</font>;<br/>
<b>consider </b><font color="Maroon" title="c23">ft1</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="E42:41"/>
( <font color="Maroon" title="c23">ft1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font> &amp; <font color="Maroon" title="c22">cP1</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="c23">ft1</font> &amp; <font color="Maroon" title="c23">ft1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>and </b><br/><a NAME="E43:41"/><i><font color="Green" title="E65">A41</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; <font color="Maroon" title="c22">cP1</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> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font> holds <br/><font color="Maroon" title="c23">ft1</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#E2:41"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="topalg_5.html#T23" target="_self" title="TOPALG_5:th.23">Th23</a></i>;<br/>
<a NAME="E44:41"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c22">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">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#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">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#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">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">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c24">x</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c24">x</font>
<b>by </b><i><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/>.= 
<font color="Maroon" title="c8">x0</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E28:41"><i><font color="Green" title="E56">A32</font></i></a>, <a class="txt" href="topalg_5.html#E1:41"><i><font color="Green" title="E39">A1</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="c22">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x</font>
<b>by </b><i><a class="ref" href="topalg_3.html#T22" title="TOPALG_3:th.22">TOPALG_3:22</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E45:41"/><b>then </b>
<font color="Maroon" title="c22">cP1</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#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</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="E46:41"/><b>then </b><i><font color="Green" title="E66">A42</font></i>: 
<font color="Maroon" title="c23">ft1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E34:41"><i><font color="Green" title="E61">A37</font></i></a>, <a class="txt" href="topalg_5.html#E36:41"><i><font color="Green" title="E62">A38</font></i></a>, <a class="txt" href="topalg_5.html#E43:41"><i><font color="Green" title="E65">A41</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c24">g1</font> = <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c13">xt</font>;<br/>
<a NAME="E47:41"/><i><font color="Green" title="E67">A43</font></i>: 
<span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c13">xt</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font>
 
<b>by </b><i><a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></i>;<br/>
<a NAME="E48:41"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c22">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c13">xt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">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><font color="Maroon" title="c22">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font> = 
<font color="Maroon" title="c8">x0</font>
<b>by </b><i><a class="ref" href="topalg_3.html#T22" title="TOPALG_3:th.22">TOPALG_3:22</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"><span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c13">xt</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E10:41"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c13">xt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E49:41"/><b>then </b>
<font color="Maroon" title="c22">cP1</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="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c13">xt</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="E50:41"/><b>then </b><i><font color="Green" title="E68">A44</font></i>: 
<font color="Maroon" title="c23">ft1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c13">xt</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E47:41"><i><font color="Green" title="E67">A43</font></i></a>, <a class="txt" href="topalg_5.html#E43:41"><i><font color="Green" title="E65">A41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">cP2</font> being  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c9">y0</font>;<br/>
<i><font color="Green" title="E69">A45</font></i>: <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">yt</font> = 
<font color="Maroon" title="c10">P</font> <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="txt" href="topalg_5.html#E5:41"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c9">y0</font>
<b>by </b><i><a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>
;<br/>
<a NAME="E53:41"/><b>then </b><i><font color="Green" title="E70">A46</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E54:41"/><b>then </b>
<font color="Maroon" title="c18">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"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">ft2</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="E56:41"/>
( <font color="Maroon" title="c26">ft2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">yt</font> &amp; <font color="Maroon" title="c25">cP2</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="c26">ft2</font> &amp; <font color="Maroon" title="c26">ft2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>and </b><br/><a NAME="E57:41"/><i><font color="Green" title="E71">A47</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; <font color="Maroon" title="c25">cP2</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> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">yt</font> holds <br/><font color="Maroon" title="c26">ft2</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 NAME="E58:41"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c25">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">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#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">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#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">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">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c27">x</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c27">x</font>
<b>by </b><i><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/>.= 
<font color="Maroon" title="c9">y0</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E28:41"><i><font color="Green" title="E56">A32</font></i></a>, <a class="txt" href="topalg_5.html#E1:41"><i><font color="Green" title="E39">A1</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="c25">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x</font>
<b>by </b><i><a class="ref" href="topalg_3.html#T22" title="TOPALG_3:th.22">TOPALG_3:22</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E59:41"/><b>then </b>
<font color="Maroon" title="c25">cP2</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#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</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="E60:41"/><b>then </b><i><font color="Green" title="E72">A48</font></i>: 
<font color="Maroon" title="c26">ft2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E37:41"><i><font color="Green" title="E63">A39</font></i></a>, <a class="txt" href="topalg_5.html#E39:41"><i><font color="Green" title="E64">A40</font></i></a>, <a class="txt" href="topalg_5.html#E57:41"><i><font color="Green" title="E71">A47</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c27">g2</font> = <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c18">yt</font>;<br/>
<a NAME="E61:41"/><i><font color="Green" title="E73">A49</font></i>: 
<span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c18">yt</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">yt</font>
 
<b>by </b><i><a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></i>;<br/>
<a NAME="E62:41"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c25">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c18">yt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">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><font color="Maroon" title="c25">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">x</font> = 
<font color="Maroon" title="c9">y0</font>
<b>by </b><i><a class="ref" href="topalg_3.html#T22" title="TOPALG_3:th.22">TOPALG_3:22</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"><span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c18">yt</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E52:41"><i><font color="Green" title="E69">A45</font></i></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c18">yt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E63:41"/><b>then </b>
<font color="Maroon" title="c25">cP2</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="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c18">yt</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="E64:41"/><b>then </b><i><font color="Green" title="E74">A50</font></i>: 
<font color="Maroon" title="c26">ft2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c18">yt</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E61:41"><i><font color="Green" title="E73">A49</font></i></a>, <a class="txt" href="topalg_5.html#E57:41"><i><font color="Green" title="E71">A47</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c28">Qt</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">G</font>;<br/>
<a NAME="E65:41"/>
 <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">G</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Maroon" title="c18">yt</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_10"><b>proof </b></a><div class="add">

<a NAME="E1:41_10"/>
for <font color="Olive" title="b1">H</font> being <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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>,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  holds   <a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
;<br/>
<b>hence </b><a NAME="E2:41_10"/>
 <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">G</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#E27:41"><i><font color="Green" title="E55">A31</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><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">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 = 
<font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="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/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</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#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c13">xt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E50:41"><i><font color="Green" title="E68">A44</font></i></a>, <a class="txt" href="topalg_5.html#E46:41"><i><font color="Green" title="E66">A42</font></i></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></i>
;<br/>

<b>thus </b><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">G</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 = 
<font color="Maroon" title="c17">G</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="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</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#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c18">yt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E64:41"><i><font color="Green" title="E74">A50</font></i></a>, <a class="txt" href="topalg_5.html#E60:41"><i><font color="Green" title="E72">A48</font></i></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c29">Qt</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">G</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Maroon" title="c18">yt</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c19">ft</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c29">Qt</font>
;<br/>

<div><i><font color="Green" title="E75">A51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c30">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/><a NAME="E1:41_11"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">s</font>,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">s</font>,0 = 
<font color="Maroon" title="c16">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">s</font>,<font color="Maroon" title="c4">j0</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E29:41"><i><font color="Green" title="E57">A33</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:41"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>
;<br/><br/><b>thus </b><a NAME="E3:41_11"/>
<font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Qt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</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>let </b><font color="Maroon" title="c31">t</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><font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Maroon" title="c31">t</font> = 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c17">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">t</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c13">xt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E50:41"><i><font color="Green" title="E68">A44</font></i></a>, <a class="txt" href="topalg_5.html#E46:41"><i><font color="Green" title="E66">A42</font></i></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></i>
;<br/><b>thus </b><font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c31">t</font> = 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c17">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">t</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c18">yt</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E64:41"><i><font color="Green" title="E74">A50</font></i></a>, <a class="txt" href="topalg_5.html#E60:41"><i><font color="Green" title="E72">A48</font></i></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E68:41"/>
<font color="Maroon" title="c17">G</font> is    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c19">ft</font>,<font color="Maroon" title="c29">Qt</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_12"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_12"/>
<font color="Maroon" title="c19">ft</font>,<font color="Maroon" title="c29">Qt</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#E27:41"><i><font color="Green" title="E55">A31</font></i></a>, <a class="txt" href="topalg_5.html#E67:41"><i><font color="Green" title="E75">A51</font></i></a>, <a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a><br/>

<b>thus </b><a NAME="E2:41_12"/>
( <font color="Maroon" title="c17">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Qt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font> &amp; <font color="Maroon" title="c17">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">yt</font> ) ) )
 <b>by </b><i><a class="txt" href="topalg_5.html#E27:41"><i><font color="Green" title="E55">A31</font></i></a>, <a class="txt" href="topalg_5.html#E67:41"><i><font color="Green" title="E75">A51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c30">G</font> = <font color="Maroon" title="c17">G</font> as    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c19">ft</font>,<font color="Maroon" title="c29">Qt</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c30">G</font>
;<br/>

<b>thus </b><a NAME="E70:41"/><i><font color="Green" title="E76">A52</font></i>: 
<font color="Maroon" title="c19">ft</font>,<font color="Maroon" title="c29">Qt</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#E27:41"><i><font color="Green" title="E55">A31</font></i></a>, <a class="txt" href="topalg_5.html#E67:41"><i><font color="Green" title="E75">A51</font></i></a>, <a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a></i>;<br/>

<b>thus </b><a NAME="E71:41"/>
<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="c30">G</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E28:41"><i><font color="Green" title="E56">A32</font></i></a></i>;<br/>

<b>thus </b><a NAME="E72:41"/>
<font color="Maroon" title="c18">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"><font color="Maroon" title="c9">y0</font></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#E53:41"><i><font color="Green" title="E70">A46</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>

<b>let </b><font color="Maroon" title="c31">F1</font> be    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c19">ft</font>,<font color="Maroon" title="c29">Qt</font>;<br/>

<b>assume </b><a NAME="E73:41"/><i><font color="Green" title="E77">A53</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="c31">F1</font>
 ;<br/>

<a NAME="E74:41"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c31">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E75:41"/><b>then </b><i><font color="Green" title="E78">A54</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E15:41"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="topalg_5.html#E14:41"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E76:41"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_13"/><i><font color="Green" title="E79">A55</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c33">x1</font>, <font color="Maroon" title="c34">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:41_13"/><i><font color="Green" title="E80">A56</font></i>: 
<font color="Maroon" title="c33">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <b>and </b><br/><a NAME="E4:41_13"/><i><font color="Green" title="E81">A57</font></i>: 
<font color="Maroon" title="c34">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E5:41_13"/><i><font color="Green" title="E82">A58</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">x1</font>,<font color="Maroon" title="c34">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E75:41"><i><font color="Green" title="E78">A54</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:41_13"/><i><font color="Green" title="E83">A59</font></i>: 
<font color="Maroon" title="c34">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:41_13"><i><font color="Green" title="E81">A57</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c32">x</font> = 
<font color="Maroon" title="c31">F1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c33">x1</font>,<font color="Maroon" title="c34">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:41_13"><i><font color="Green" title="E82">A58</font></i></a>, <a class="txt" href="topalg_5.html#E1:41_13"><i><font color="Green" title="E79">A55</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c19">ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c33">x1</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E70:41"><i><font color="Green" title="E76">A52</font></i></a>, <a class="txt" href="topalg_5.html#E3:41_13"><i><font color="Green" title="E80">A56</font></i></a>, <a class="txt" href="topalg_5.html#E6:41_13"><i><font color="Green" title="E83">A59</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="c16">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c33">x1</font>,<font color="Maroon" title="c34">x2</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:41_13"><i><font color="Green" title="E80">A56</font></i></a>, <a class="txt" href="topalg_5.html#E4:41_13"><i><font color="Green" title="E81">A57</font></i></a>, <a class="txt" href="topalg_5.html#E13:41"><i><font color="Green" title="E47">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c16">Ft</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c32">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:41_13"><i><font color="Green" title="E82">A58</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E77:41"/><b>then </b>
<font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Ft</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E75:41"><i><font color="Green" title="E78">A54</font></i></a>, <a class="txt" href="topalg_5.html#E17:41"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E78:41"/>
<font color="Maroon" title="c30">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c31">F1</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E73:41"><i><font color="Green" title="E77">A53</font></i></a>, <a class="txt" href="topalg_5.html#E30:41"><i><font color="Green" title="E58">A34</font></i></a></i>;<br/>


</div>
